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\def\Ker{\operatorname{Ker}}
\def\M{\mathsf{M}}
\def\K{\mathsf{K}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\def\T{\mathsf{T}}
\def\F{\mathcal{F}}
\def\aa{\boldsymbol{a}}
\def\bb{\boldsymbol{b}}
\def\cc{\boldsymbol{c}}
\def\ee{\boldsymbol{e}}
\def\qq{\boldsymbol{q}}
\def\rr{\boldsymbol{r}}
\def\ss{\boldsymbol{s}}
\def\uu{\boldsymbol{u}}
\def\vv{\boldsymbol{v}}
\def\xx{\boldsymbol{x}}
\def\yy{\boldsymbol{y}}
\def\zz{\boldsymbol{z}}
\def\llambda{\boldsymbol{\lambda}}
\def\aalpha{\boldsymbol{\alpha}}

\def\zero{\boldsymbol{0}}
\def\supp{\operatorname{supp}}
\def\conv{\operatorname{conv}}
\def\ds{\displaystyle}
\def\KG{\operatorname{KG}}
\def\cd{\operatorname{cd}}
\def\rk{\operatorname{rk}}
\def\alt{\operatorname{alt}}
%\def\H{\mathcal{H}}
\def\id{\operatorname{id}}
\def\diam{\operatorname{diam}}
\def\stab{\operatorname{stab}}
\def\sd{\operatorname{sd}}
\def\mod{\operatorname{mod}}
\def\coind{\operatorname{coind}}

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\title{\bf Colorful Subhypergraphs in\\ Uniform Hypergraphs}

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\author{Meysam Alishahi\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Shahrood University of Technology, Iran\\
\small\tt meysam\_alishahi@shahroodut.ac.ir\\
}

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\date{\dateline{May 22, 2016}{Jan 25, 2017}{Feb 3, 2017}\\
\small Mathematics Subject Classifications: 05C15}

\begin{document}

\maketitle

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\begin{abstract}
There are several topological results ensuring in any properly colored 
graph the existence of 
a colorful complete bipartite subgraph, whose order is bounded from below by some 
topological invariants of some topological spaces associated to the graph.  
Meunier [\textit{Electron. J. Combin.,} 2014] presented the first colorful type result for uniform hypergraphs. 
In this paper, we give some new generalizations of the $\mathbb{Z}_p$-Tucker~Lemma 
and by 
use of them, we improve Meunier's result and some other colorful results by Simonyi, Tardif, 
and Zsb{\'{a}}n [\textit{Electron. J. Combin.,} 2014] and by  Simonyi and  Tardos  
[\textit{Electron. J. Combin.,} 2007] 
to uniform hypergraphs.  Also, we introduce some new lower bounds for the chromatic 
number and local chromatic number of uniform hypergraphs. A hierarchy between these 
lower bounds is presented as well.

  % keywords are optional
\bigskip\noindent \textbf{Keywords:} chromatic number of hypergraphs, 
$\mathbb{Z}_p$-Tucker-Ky~Fan lemma, colorful complete hypergraph, 
$\mathbb{Z}_p$-box-complex, $\mathbb{Z}_p$-Hom-complex. 
\end{abstract}


\section{\bf Introduction}
\subsection{{\bf Background and Motivations}}
In 1955, Kneser~\cite{MR0068536} posed a conjecture about the chromatic number 
of Kneser graphs. In 1978, Lov{\'a}sz~\cite{MR514625} proved this conjecture by 
using algebraic topology.
The Lov{\'a}sz proof marked the beginning of the history of topological combinatorics.
Nowadays,  it is an active stream of research to study the coloring properties of graphs  
by using algebraic topology. There are several lower bounds for the chromatic number 
of graphs  related to the indices of some topological spaces defined based on the 
structure of graphs. 
However, for hypergraphs, there are few such lower bounds, 
see~\cite{2013arXiv1302.5394A,MR953021,Iriye20131333,MR1081939,MR2279672}.


A {\it hypergraph} ${\mathcal H}$ is a pair $(V({\mathcal H}),E({\mathcal H}))$, 
where $V({\mathcal H})$ is a finite nonempty set, called the vertex set 
of ${\mathcal H}$, and $E({\mathcal H})$ is a family of nonempty subsets of 
$V({\mathcal H})$, called the edge set of ${\mathcal H}$. 
Throughout the paper, by a nonempty hypergraph, we mean a hypergraph 
with at least one edge. The number of vertices of a hypergraph is called its {\it order}. 
If any edge $e\in E({\mathcal H})$ has cardinality $r$, then the hypergraph 
${\mathcal H}$ is called  {\it $r$-uniform.} For a set $U\subseteq V({\mathcal H})$, 
the {\it induced subhypergraph by $U$,} denoted ${\mathcal H}[U]$, is a hypergraph 
with the vertex set $U$ and the edge set 
$\{e\in E({\mathcal H}):\; e\subseteq U\}$.
Throughout the paper, by a {\it graph,} we mean a $2$-uniform hypergraph. 
Let $r$ and $q$ be a positive integers, where $q\geq r\geq 2$. 
An $r$-uniform hypergraph ${\mathcal H}$ is called {\it $q$-partite}
if its vertex set can be partitioned into 
subsets $U_1,\ldots,U_q$ such that  each edge of ${\mathcal H}$ intersects each 
part $U_i$ in at most one vertex.  
A {\it complete $r$-uniform $q$-partite hypergraph} is an $r$-uniform $q$-partite hypergraph containing all possible edges. 
%If such a hypergraph ${\mathcal H}$ contains all possible edges, then we call it a {\it complete $r$-uniform $q$-partite hypergraph.} 
Also, the hypergraph  ${\mathcal H}$ is said to be {\it balanced}  
if the values of the $|U_j|$'s  for 
$j =1,\ldots,q$ differ by at most one, i.e., $|U_i|-|U_j|\leq 1$ for each $i,j\in[q]$.

Let ${\mathcal H}$ be an $r$-uniform hypergraph and $U_1,\ldots, U_q$ be $q$ pairwise disjoint 
subsets of $V({\mathcal H})$.
The hypergraph ${\mathcal H}[U_1,\ldots, U_q]$ is a subhypergraph of ${\mathcal H}$ with the vertex set $\cup_{i=1}^q U_i$ and the edge set $$E({\mathcal H}[U_1,\ldots, U_q])=\left\{e\in E({\mathcal H}):\; e\subseteq \displaystyle\bigcup_{i=1}^q U_i\mbox{ and } |e\cap U_i|\leq 1\mbox{ for each } i\in[q]\right\}.$$
Note that ${\mathcal H}[U_1,\ldots, U_q]$ is an $r$-uniform $q$-partite hypergraph with parts $U_1,\ldots,U_q$.
By the symbols $[n]$ and ${[n]\choose r}$, we respectively mean the set $\{1,\ldots,n\}$ and the  family of all $r$-subsets of $[n]$. 
The hypergraph $K^r_n=\left([n],{[n]\choose r}\right)$ is called  the {\it complete $r$-uniform hypergraph} with $n$ vertices. For $r=2$, we would rather use $K_n$ instead of $K_n^2$.
The largest possible integer $n$ such that ${\mathcal H}$ 
contains $K^r_n$ as a  subhypergraph 
is called the {\it clique number of ${\mathcal H}$}, denoted $\omega({\mathcal H})$.

Let $L$ be a positive integer. 
A {\it proper $L$-coloring} of a hypergraph ${\mathcal H}$ is a map $c:V({\mathcal H})\longrightarrow [L]$ such that $\mathcal{H}$ has no monochromatic edge, that is,
there is no edge $e\in E(\mathcal{H})$ with $|c(e)|=1$. 
A hypergraph ${\mathcal H}$ is called $L$-colorable if it admits a proper 
$L$-coloring.  
The minimum possible $L$ for which ${\mathcal H}$ is $L$-colorable is called {\it the chromatic number of ${\mathcal H}$}, denoted $\chi({\mathcal H})$. If there is no such an $L$, we define the chromatic number to be infinite.
Let $c$ be a proper coloring  of ${\mathcal H}$ and  $U_1,\ldots, U_q$ be 
$q$ pairwise disjoint subsets  of $V({\mathcal H})$.
The hypergraph ${\mathcal H}[U_1,\ldots, U_q]$ is said to be {\it colorful } if  
for each $j\in[q]$, the vertices in $U_j$ get pairwise distinct colors.
For a properly colored graph $G$, a subgraph is called {\it multicolored} if its vertices 
get pairwise distinct colors.

For a hypergraph ${\mathcal H}$, {\it the Kneser hypergraph} ${\rm KG}^r({\mathcal H})$ is 
an $r$-uniform hypergraph with the vertex set $E({\mathcal H})$ and whose edges are 
formed by $r$ pairwise vertex-disjoint edges of ${\mathcal H}$, i.e.,
$$E({\rm KG}^r({\mathcal H}))=\left\{ \{e_1,\ldots,e_r\}:\; e_i\cap e_j=
\varnothing\mbox{ for each } i\neq j\in[r] \right\}.$$  
The Kneser hypergraph ${\rm KG}^r\left(K_n^k\right)$ is called the ``usual" 
Kneser hypergraph, which is denoted by ${\rm KG}^r(n,k)$. 
Coloring properties of Kneser hypergraphs $\KG^r(n,k)$ have been extensively studied  
in the literature. Lov\'asz~\cite{MR514625} (for $r=2$) and Alon, Frankl, 
and Lov\'asz~\cite{MR857448} determined the chromatic number of ${\rm KG}^r(n,k)$. 
For an integer $r\geq 2$, they proved that
$$\chi\left({\rm KG}^r(n,k)\right)= \left\lceil{n-r(k-1)\over r-1}\right\rceil.$$
For a hypergraph ${\mathcal H}$, {\it the $r$-colorability defect of ${\mathcal H}$,} 
denoted ${\rm cd}_r({\mathcal H})$, is the 
minimum number of vertices that should be removed so that the 
induced subhypergraph by the remaining vertices is $r$-colorable, i.e., 
$${\rm cd}_r({\mathcal H})=\min\left\{|U|:\; {\mathcal H}[V({\mathcal H})\setminus U]\mbox{ is $r$-colorable}\right\}.$$
For a hypergraph ${\mathcal H}$, 
Dol'nikov~\cite{MR953021}~(for $r=2$) and K{\v{r}}{\'{\i}}{\v{z}}~{\rm \cite{MR1081939} proved that 
$$\chi({\rm KG}^r({\mathcal H}))\geq \left\lceil{{\rm cd}_r({\mathcal H})\over r-1}\right\rceil,$$
which is a generalization of the results by Lov\'asz~\cite{MR514625} and Alon, Frankl and Lov\'asz~\cite{MR857448}.

For a positive integer $r$, let $\mathbb{Z}_r=\{\omega,\omega^2\ldots,\omega^r\}$ 
be a cyclic group of order $r$ with generator $\omega$.
Consider a vector  $X=(x_1,x_2,\ldots,x_n)\in(\mathbb{Z}_r\cup\{0\})^n$.
{\it An alternating 
subsequence of $X$} is a sequence $x_{i_1},x_{i_2},\ldots,x_{i_m}$ 
of nonzero terms of $X$ such that $i_1<\cdots<i_m$ and $x_{i_j}\neq x_{i_{j+1}}$ for each $j\in [m-1]$. 
We denote by ${\rm alt}(x)$ the maximum possible length of an 
alternating subsequence of $X$.
For a vector $X=(x_1,x_2,\ldots,x_n)\in (\mathbb{Z}_r\cup\{0\})^n$ and for an $\varepsilon\in\Z_p$, set 
$X^\varepsilon=\{i\in[n]:\; x_i=\varepsilon\}$.
Note that, by abuse of notation, we can write $X=(X^\varepsilon)_{\varepsilon\in \mathbb{Z}_r}$.
For two vectors $X,Y\in (\mathbb{Z}_r\cup\{0\})^n$, by $X\subseteq Y$, we mean
$X^\varepsilon\subseteq Y^\varepsilon$ for each $\varepsilon\in\mathbb{Z}_r$.

For a hypergraph ${\mathcal H}$ and a bijection $\sigma:[n]\longrightarrow V({\mathcal H})$, define 
$${\rm alt}_r({\mathcal H},\sigma)=\ds\max\left\{{\rm alt}(X): X\in (\mathbb{Z}_r\cup\{0\})^n\mbox{ s.t. } E({\mathcal H}[\sigma(X^\varepsilon)])=\varnothing\mbox{ for each  } \varepsilon\in\mathbb{Z}_r \right\}.$$
Also, let 
$${\rm alt}_r({\mathcal H})=\displaystyle\min_{\sigma} {\rm alt}_r({\mathcal H},\sigma),$$
where the minimum is taken over all bijections $\sigma:[n]\longrightarrow V({\mathcal H})$.
One can readily check that for any hypergraph ${\mathcal H}$, we have $|V({\mathcal H})|-{\rm alt}_r({\mathcal H})\geq {\rm cd}_r({\mathcal H})$ and the inequality is often strict, see~\cite{2013arXiv1302.5394A}. 
The present author and Hajiabolhassan~\cite{2013arXiv1302.5394A} improved  Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}} result by proving that 
for any hypergraph ${\mathcal H}$ and for any integer $r\geq 2$, the quantity $\left\lceil{ |V({\mathcal H})|-{\rm alt}_r({\mathcal H})\over r-1}\right\rceil$ is a lower bound for the chromatic number of 
${\rm KG}^r({\mathcal H})$.  
%Note that since $|V({\mathcal H})|-{\rm alt}_r({\mathcal H})\geq {\rm cd}_r({\mathcal H})$, it improves Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}}} lower bound.
%{\color{blue}Using this lower bound, the chromatic number of some families of graphs and hypergraphs are computed, see~\cite{2014arXiv1401.0138A,2014arXiv1403.4404A,2014arXiv1407.8035A,2015arXiv150708456A,2013arXiv1302.5394A,HaMe16}. } 
There are some other lower bounds for the chromatic number of graphs, being sometimes 
better than the former discussed lower bounds, for instance, see~\cite{2016arXiv160708780A,SiTaZs13,MR2279672}.
They are based on some topological  indices of some topological spaces 
connected to the structure of graphs.
In spite of these lower bounds being better, they are~not combinatorial and most of the 
times they are difficult to compute.

The existence of large colorful bipartite subgraphs in a properly colored graph has 
been extensively studied in the 
literature, 
see~\cite{2013arXiv1302.5394A,MR2971704,MR2763055,
SiTaZs13,MR2279672,MR2351519}. 
To be more specific, there are several theorems ensuring the existence of a colorful bipartite 
subgraph in any properly colored graph, whose order is bounded 
form below by  a function of 
some topological parameters connected to the graph. 
%such that the bipartite subgraph has a specific number of vertices related to some topological parameters connected to the graph. 
In this regard, 
Simonyi and Tardos~\cite{MR2351519} improved  Dol'nikov's lower bound and proved that 
in any proper coloring of a Kneser graph ${\rm KG}^2({\mathcal H})$, there is a multicolored complete bipartite graph 
$K_{\left\lceil{{\rm cd}_2({\mathcal H})\over 2}\right\rceil,\left\lfloor{{\rm cd}_2({\mathcal H})\over 2}\right\rfloor}$ 
such that the ${\rm cd}_2({\mathcal H})$ different colors occur alternating on the 
two parts of the bipartite graph with respect to their natural order. 
By a combinatorial proof, the present author and Hajiabolhassan~\cite{2013arXiv1302.5394A}  improved this result. They proved that the result 
remains true if we replace ${\rm cd}_2({\mathcal H})$ by $|V(\mathcal{H})|-{\rm alt}_2({\mathcal H})$. Also, 
a stronger result is proved by Simonyi, Tardif, and  Zsb{\'{a}}n~\cite{SiTaZs13}.
\begin{theorem}{\rm (Zig-zag Theorem~\cite{SiTaZs13}).}\label{zigzag}
Let $G$ be a nonempty graph, 
which is properly colored with arbitrary number of colors. 
Then $G$ contains a multicolored complete bipartite subgraph $K_{\lceil{t\over2}\rceil,\lfloor{t\over2}\rfloor}$,
where ${\rm Xind}({\rm Hom}(K_2,G))+2= t$. Moreover, colors appear alternating on the two sides of the bipartite subgraph with respect to their natural ordering.
\end{theorem}

%They proved that for any proper coloring of a graph $H$,  there is a colorful bipartite subgraph $K_{\lceil{t\over2}\rceil,\lfloor{t\over2}\rfloor}$, where ${\rm Xind}({\rm Hom}(K_2,H))+2= t$. Moreover, colors appear alternating  on the two sides of the bipartite subgraph with respect to their natural ordering. 
The quantity ${\rm Xind}({\rm Hom}(K_2,G))$ appearing in the statement of previous theorem is the cross-index of  the 
Hom-complex ${\rm Hom}(K_2,G)$,  which will be defined in 
Subsection~\ref{Boxdefin}. We should mention that there are some other weaker similar 
results in terms of some other topological parameters, see~\cite{MR2279672,MR2351519}.
 
Note that all the prior mentioned results concern the existence of colorful bipartite subgraphs 
in properly colored graphs ($2$-uniform hypergraphs).
In 2014, Meunier~\cite{Meunier14} 
found the first colorful type result for uniform hypergraphs, see Theorem~\ref{colorfulhyper}.
He proved that for any prime number $p$, any properly colored Kneser 
hypergraph $\KG^p(\mathcal{H})$ must contain a colorful
balanced complete $p$-uniform $p$-partite subhypergraph,  
whose order is bounded from below by $|V(\mathcal{H})|-\alt_p(\mathcal{H})$, see Theorem~\ref{colorfulhyper}.


\subsection{\bf Main Results}
For a given graph $G$, 
there are several complexes defined based on the structure of $G$,  
for instance, the box-complex of $G$, denoted ${\rm B}_0(G)$,  and the Hom-complex of $G$, denoted ${\rm Hom}(K_2,G)$, see~\cite{MR1988723,SiTaZs13,MR2279672}. 
%Also, there are some lower bounds for the chromatic number of graphs related to some indices of these complexes~\cite{SiTaZs13,MR2279672}.
In this paper, we naturally generalize the  definitions of  box-complex and Hom-complex 
of graphs to uniform hypergraphs. 
Also, the definition of $\mathbb{Z}_p$-cross-index of $\Z_p$-posets will be introduced.
Using these complexes, 
as a first main result of this paper, we generalize Meunier's theorem~\cite{Meunier14} (Theorem~\ref{colorfulhyper})
to the following theorem.
\begin{theorem}\label{maincolorfulindex}
Let $r$ and $p$ be two positive integers, where $p$ is prime and $p\geq r\geq 2$.
Also, let ${\mathcal H}$ be an $r$-uniform hypergraph and $c:V({\mathcal H})\longrightarrow[L]$ be a proper coloring of 
${\mathcal H}$ {\rm (}$L$ arbitrary{\rm )}. Then we have the 
following assertions.
\begin{itemize}
\item[{\rm (i)}] There is a colorful balanced complete $r$-uniform $p$-partite 
subhypergraph in  ${\mathcal H}$
with ${\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1$ vertices.
In particular,
$$\chi({\mathcal H})\geq {{\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1\over r-1}.$$
\item[{\rm (ii)}] If $p\leq \omega({\mathcal H})$, then there is a colorful balanced complete 
$r$-uniform $p$-partite subhypergraph in ${\mathcal H}$ with 
${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p$ vertices.
In particular,
$$\chi({\mathcal H})\geq {{\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p\over r-1}.$$
\end{itemize}
\end{theorem}

The quantities ${\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))$ and 
${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))$ appearing in the statement 
of Theorem~\ref{maincolorfulindex} are respectively
the $\mathbb{Z}_p$-index and  the $\mathbb{Z}_p$-cross-index 
of the $\mathbb{Z}_p$-box-complex ${\rm B_0}({\mathcal H},\mathbb{Z}_p)$  
and the $\mathbb{Z}_p$-Hom-complex ${\rm Hom}(K^r_p,{\mathcal H})$, which will be 
defined in Subsection~\ref{Boxdefin}. 
It is worth mentioning that these quantities are defined in a way that if $p=2$, then 
for any graph $G$, we have
${\rm B_0}(G)={\rm B_0}(G,\mathbb{Z}_2)$,  
${\rm Hom}_{\Z_2}(K_2,G)={\rm Hom}(K_2,G)$, 
${\rm ind}(-)={\rm ind}_{\mathbb{Z}_2}(-)$, and 
${\rm Xind}(-)={\rm Xind}_{\mathbb{Z}_2}(-)$.
In other words, they are true generalizations of box-complex, Hom-complex  and 
their indices to the case of hypergraphs. The assumption $p\leq \omega({\mathcal H})$ 
in the statement of Theorem~\ref{maincolorfulindex} 
is required to grantee that the ground set of the $\mathbb{Z}_p$-Hom-complex 
${\rm Hom}(K^r_p,{\mathcal H})$ is~not empty, see Section~\ref{Boxdefin}. 

Let $\mathcal{H}$ be a properly colored $r$-uniform hypergraph with $L$ colors. 
Clearly, the existence of a colorful balanced complete $p$-partite 
subhypergraph ${\mathcal H}[U_1,\ldots,U_p]$ in  ${\mathcal H}$ 
provides a lower bound on $L$.
To see this, note that any color appears in at most $r-1$ vertices of each edge of 
$\mathcal{H}$ and consequently, in at most $r-1$ number of $U_i$'s. Clearly, this implies 
$$L\geq {1\over r-1}\sum\limits_{i=1}^p|V_i|.$$
This observation shows that the inequalities appearing in the statement 
of Theorem~\ref{maincolorfulindex} are immediate consequences of the existence 
of the claimed subhypergraphs.  Therefore, for the proof of this theorem, we just need 
to prove the existence of such subhypergraphs. 


As we said before, there are several topological lower bounds for the chromatic number of graphs. The following chain of inequalities provides a hierarchy between some of these lower bounds;  
%For a nonempty graph $G$ and for $p=2$, it is proved~\cite{2014arXiv1403.4404A,2013arXiv1302.5394A,SiTaZs13,MR2279672} that
\begin{equation}\label{equation}
\begin{array}{lll}
\chi(G) &\geq & {\rm Xind}({\rm Hom}(K_2,G))+2  \geq  {\rm ind}({\rm B_0}(G))+1\\
&\geq &  {\rm coind}({\rm B_0}(G))+1\geq  |V({\mathcal F})|-{\rm alt}_2({\mathcal F})
\geq  {\rm cd}_2({\mathcal F}),
\end{array}
\end{equation} 
where ${\mathcal F}$ is any hypergraph such that ${\rm KG}^2({\mathcal F})$ and $G$ 
are isomorphic, see~\cite{2016arXiv160708780A,2013arXiv1302.5394A,SiTaZs13,MR2279672}. 
It should be mentioned that for any graph $G$, there are several 
hypergraphs ${\mathcal F}$ such that ${\rm KG}^2({\mathcal F})$ and $G$ are isomorphic.  
In the next theorem, we compare the lower bounds for the chromatic number of 
$r$-uniform hypergraphs introduced in Theorem~\ref{maincolorfulindex}, providing a 
hierarchy between these lower bounds.
Note that, in view of the discussion right after Theorem~\ref{maincolorfulindex}, 
next theorem somehow generalizes Chain~\ref{equation} to the case of hypergraphs. 
\begin{theorem}\label{inequalities}
Let $r$ be a positive integer and $p$ be a prime number, where $2\leq r\leq p$.
For any $r$-uniform hypergraph ${\mathcal H}$, we have the followings.\\  
{\rm (i)} If $p\leq \omega({\mathcal H})$, then
$${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p \geq  {\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1.$$
{\rm (ii)} If ${\mathcal H}={\rm KG}^r({\mathcal F})$ for some hypergraph ${\mathcal F}$, then 
$$
 {\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1
\geq   |V({\mathcal F})|-{\rm alt}_p({\mathcal F})
\geq  {\rm cd}_p({\mathcal F}).
$$
\end{theorem}


In view of  Theorem~\ref{inequalities}, Theorem~\ref{maincolorfulindex} is a common 
extension of Theorem~\ref{zigzag} and Theorem~\ref{colorfulhyper} 
(Meunier's colorful theorem). Furthermore, for $r=2$, Theorem~\ref{maincolorfulindex} 
implies the next corollary, which is a generalization of Theorem~\ref{zigzag}.
\begin{corollary}\label{cor1}
Let $p$ be a prime number and $G$ be a nonempty graph, which is properly colored with arbitrary number of colors. Then  there is a multicolored 
complete $p$-partite subgraph $K_{n_1,n_2,\ldots,n_p}$ of $G$ such that 
\begin{itemize}
\item $\ds\sum_{i=1}^pn_i={\rm ind}_{\mathbb{Z}_p}({\rm B_0}(G,\mathbb{Z}_p))+1$,
\item $|n_i-n_j|\leq 1$ for each $i,j\in[p]$. 
\end{itemize}
Moreover, if $p\leq \omega(G)$, then ${\rm ind}_{\mathbb{Z}_p}({\rm B_0}(G,\mathbb{Z}_p))+1$ 
can be replaced by ${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K_p,G))+p$.
\end{corollary}

In view of the prior mentioned results, the following question naturally arises.
\begin{question} Do Theorem~\ref{maincolorfulindex} and Theorem~\ref{inequalities} remain true for non-prime $p$?
\end{question}


\subsection{\bf Applications to Local Chromatic Number of Uniform Hypergraphs}
For a graph $G$ and a vertex $v\in V(G)$, the {\it closed neighborhood of $v$,} denoted $N[v]$, is the set $\{v\}\cup\{u:\; uv\in E(G)\}$. The {\it local chromatic number of $G$,} denoted 
$\chi_l(G)$, is defined in~\cite{ERDOS198621} as follows:
$$\chi_l(G)=\ds\min_c\max\{|c(N[v])|:\; v\in V(G)\},$$
where the minimum is taken over all proper colorings $c$ of $G$.
Theorem~\ref{zigzag} gives the following lower bound for the local chromatic 
number of a nonempty graph $G$:  
\begin{equation}\label{localloerzigzag}
\chi_l(G)\geq \left\lceil{{\rm Xind}({\rm Hom}(K_2,G))\over 2}\right\rceil+2.
\end{equation}
Note that for a Kneser hypergraph $\KG^2({\mathcal H})$,
by using the Simonyi--Tardos colorful result~\cite{MR2351519} or its extension given by 
the present author and Hajiabolhassan~\cite{2013arXiv1302.5394A}, 
there are two similar lower 
bounds for $\chi_l(\KG^2({\mathcal H}))$, which respectively use $\cd_2({\mathcal H})-2$ and 
$|V({\mathcal H})|-\alt_2({\mathcal H})-2$ instead of ${\rm Xind}({\rm Hom}(K_2,G))$ in Inequality~\ref{localloerzigzag}. 
However, as it is stated in Theorem~\ref{inequalities}, the lower bound in terms 
of ${\rm Xind}({\rm Hom}(K_2,G))+2$ is better than these two last mentioned lower bounds.
Using Corollary~\ref{cor1}, we have the following lower bound for the local 
chromatic number of graphs, which is an improvement of Inequality~\ref{localloerzigzag}. 
\begin{corollary}\label{locallowerp}
Let $G$ be a nonempty graph and  $p$ be a prime number. 
Then $$\chi_l(G)\geq t-\left\lfloor{t\over p}\right\rfloor+1,$$ where 
$t={\rm ind}_{\mathbb{Z}_p}({\rm B_0}(G,\mathbb{Z}_p))+1$.
Moreover, if $p\leq \omega(G)$, then ${\rm ind}_{\mathbb{Z}_p}({\rm B_0}(G,\mathbb{Z}_p))+1$ 
can be replaced with ${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K_p,G))+p$.
\end{corollary}
Note that if we set $p=2$, then previous theorem implies the Simonyi--Tardos~lower bound 
for the local chromatic number. Moreover, in general, 
this lower bound might be even better than the Simonyi--Tardos~lower bound. To see this, let $k$ be a fixed integer, where $k\geq 2$. 
Consider the Kneser graph $\KG^2(n,k)$ and let $p(n)$ be a prime number such that
$p=p(n)\in O(\ln n)$.  By Theorem~\ref{inequalities},  for $n\geq pk$, we have  
$${\rm ind}_{\mathbb{Z}_p}({\rm B_0}(\KG^2(n,k),\mathbb{Z}_p))+1\geq \cd_p(K_n^k)=n-p(k-1).$$
Note that
the lower bound for $\chi_l(\KG^2(n,k))$ coming form Inequality~\ref{localloerzigzag}
is \begin{equation}\label{equation2}
1+\left\lceil{n-2k+2\over 2}\right\rceil={n\over 2}-o(1),
\end{equation} 
while, 
in view of Corollary~\ref{locallowerp}, we have
$$\chi_l(\KG^2(n,k))\geq n-p(k-1)-\left\lfloor{n-p(k-1)\over p}\right\rfloor+1=n-o(n),$$
which is clearly better than the quantity in Equation~\ref{equation2} provided 
that $n$ is sufficiently large. 
However, since the induced subgraph by the neighbors of any vertex of $\KG(n,k)$ is isomorphic to
$\KG(n-k,k)$, we have $\chi_l(\KG(n,k))\geq n-3(k-1),$ which is better than the 
above-mentioned lower bounds. 

Before stating the next result, we remind the reader that for a hypergraph $\mathcal{H}$, 
the maximum number of vertices of $\mathcal{H}$ containing no edge is called 
{\it the independence number of $\mathcal{H}$}, which is denoted by $\alpha(\mathcal{H})$. 
\begin{corollary}
Let $\F$ be a hypergraph and $\alpha(\F)$ be its independence number. Then for any prime number $p$, we have
$$\chi_l(\KG^2(\F))\geq \left\lceil{(p-1)|V(\F)|\over p}\right\rceil-(p-1)\cdot\alpha(\F)+1.$$
\end{corollary}
\begin{proof}
In view of Theorem~\ref{inequalities}, we have  
$${\rm ind}_{\mathbb{Z}_p}({\rm B_0}(\KG^2(\F),\mathbb{Z}_p))+1\geq \cd_p(\F)\geq |V(\F)|-p\cdot\alpha(\F).$$
Now, Corollary~\ref{locallowerp} implies the assertion.
\end{proof}
Meunier~\cite{Meunier14} naturally generalized the definition of 
local chromatic number of graphs 
to  uniform hypergraphs as follows. Let ${\mathcal H}$ be a uniform hypergraph. 
For a 
set $X\subseteq V({\mathcal H})$, the neighborhood of $X$, denoted $\mathcal{N}(X)$, 
is defined as follows; 
$${\mathcal N}(X)=\{v\in V({\mathcal H}):\; \exists\; e\in E({\mathcal H})\mbox{ such that } e\setminus X=\{v\}\}.$$ 
The {\it closed neighborhood of $X$,} denoted $\mathcal{N}[X]$, is the set 
$X\cup {\mathcal N}(X)$. 
The {\it local chromatic number of ${\mathcal H}$} is defined as follows: 
$$\chi_l({\mathcal H})=\ds\min_c\max\{|c({\mathcal N}[e\setminus \{v\}])|:\; 
e\in E({\mathcal H})\mbox{ and } v\in e\},$$
where the minimum is taken over all proper colorings $c$ of ${\mathcal H}$.

Meunier~\cite{Meunier14}, by using his colorful theorem (Theorem~\ref{colorfulhyper}), 
generalized the Simonyi--Tardos~Lower bound~\cite{MR2351519}  for the local chromatic 
number of Kneser graphs to the local chromatic number of Kneser hypergraphs. He proved that  for any hypergraph $\mathcal{F}$ and for any prime number $p$, we have 
$$
\chi_l(\KG^p(\mathcal{F}))\geq \min\left(\left\lceil{|V(\mathcal{F})|-\alt_p(\mathcal{F})\over p}\right\rceil+1, \left\lceil{|V(\mathcal{F})|-\alt_p(\mathcal{F})\over p-1}\right\rceil  \right).$$ 
In what follows, we generalize this result.
\begin{theorem}\label{localhyper}
Let ${\mathcal H}$ be a nonempty $r$-uniform hypergraph and $p$ be a prime number, 
where $2\leq r\leq p\leq \omega({\mathcal H})$. Let $t={\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p$.
If  $t=ap+b$, where $a$ and $b$ are nonnegative integers and
$0\leq b\leq p-1$, then
$$\chi_l({\mathcal H})\geq  \min\left( \left\lceil{(p-r+1)a+\min(p-r+1,b)\over \min(p-r+1,r-1)}\right\rceil+1, \left\lceil{t\over r-1}\right\rceil  \right).$$ 
\end{theorem}
\begin{proof}
Let $c$ be an arbitrary proper coloring of ${\mathcal H}$ and
let ${\mathcal H}[U_1,\ldots,U_p]$ be the balanced colorful complete $r$-uniform $p$-partite 
subhypergraph of ${\mathcal H}$, whose existence is ensured by Part~(ii) of 
Theorem~\ref{maincolorfulindex}. 
Note that $b$ of the subsets $U_i$'s, say $U_1,\ldots,U_b$, have the cardinality 
$\lceil{t\over p}\rceil$, while the others 
have the cardinality $\lfloor{t\over p}\rfloor\geq 1$.
Consider $U_1,\ldots,U_{p-r+1}$. 
Since any color appears  in at most $r-1$ vertices in $\ds\bigcup_{i=1}^p U_i$, 
we have
$\left|\ds\bigcup_{i=1}^p c(U_i)\right|\geq \left\lceil{t\over r-1}\right\rceil$. 
Two different cases will be distinguished. 
\begin{enumerate} 
\item  If $\left|\ds\bigcup_{i=1}^{p-r+1} c(U_i)\right|<\left\lceil{t\over r-1}\right\rceil$,
then there is a vertex $v\in \ds\bigcup_{i=p-r+2}^p U_i$, whose color is~not in $\ds\bigcup_{i=1}^{p-r+1} c(U_i)$. Consider an edge $e$ of ${\mathcal H}[U_1,\ldots,U_p]$ containing $v$
and such that $e\cap U_i=\varnothing$ for $i=1,\ldots, p-r$ and 
$|e\cap U_i|=1$ for $i=p-r+1,\ldots,p$. 
Let $e\cap U_{p-r+1}=\{u\}$.
Clearly, 
since $v\in e\setminus \{u\}$ and 
$\bigcup\limits_{i=1}^{p-r+1} U_i\subseteq {\mathcal N}(e\setminus \{u\})$, 
we have 
$$\{v\}\cup\ds\left(\bigcup_{i=1}^{p-r+1} U_i\right)\subseteq (e\setminus \{u\})
\cup {\mathcal N}(e\setminus \{u\})={\mathcal N}[e\setminus \{u\}].$$ 
Note that 
since any color appears in at most $\min(p-r+1,r-1)$ vertices in  $\ds\bigcup_{i=1}^{p-r+1} U_i$, we have 
$$\left|\bigcup_{i=1}^{p-r+1} c(U_i)\right|\geq \ds\left\lceil{\sum_{i=1}^{p-r+1}|U_i|\over \min(p-r+1,r-1)}\right\rceil.$$
On the other hand, since $c(v)\not\in \ds\bigcup_{i=1}^{p-r+1} c(U_i)$ and $\ds\sum_{i=1}^{p-r+1}|U_i| =(p-r+1)a+\min(p-r+1,b)$, we have   
$$\begin{array}{lll}
|c({\mathcal N}[e\setminus \{u\}])|& \geq & \left|\{c(v)\}\cup\ds\left(\bigcup_{i=1}^{p-r+1} c(U_i)\right)\right|\\ \\
   						   & \geq & 1+ \ds\left\lceil{\sum_{i=1}^{p-r+1}|U_i|\over\min(p-r+1, r-1)}\right\rceil\\ \\ 
						   &    =   & 1+ \ds\left\lceil{(p-r+1)a+\min(p-r+1,b)\over\min(p-r+1, r-1)}\right\rceil,\\ \\ 
\end{array}
$$ 
which completes the proof in Case~1. 
\item If $\left|\ds\bigcup_{i=1}^{p-r+1} c(U_i)\right|\geq\left\lceil{t\over r-1}\right\rceil$,
then consider an edge $e$ of ${\mathcal H}[U_1,\ldots,U_p]$ such that 
$e\cap U_i=\varnothing$ for $i=1,\ldots, p-r$ and 
$|e\cap U_i|=1$ for $i=p-r+1,\ldots,p$. 
%$|e\cap U_{p-r+1}|=1$ and  
Let $e\cap U_{p-r+1}=\{u\}$.
One can see that 
$$\ds\bigcup_{i=1}^{p-r+1} c(U_i)\subseteq c({\mathcal N}(e\setminus \{u\})),$$
which completes the proof in Case~2.\qedhere
\end{enumerate}
\end{proof}

\begin{corollary}
Let ${\mathcal H}$ be a nonempty $p$-uniform hypergraph, 
where $p$ is a prime number.
Then
$$\chi_l({\mathcal H})\geq \min\left(\left\lceil{{\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^p_p,{\mathcal H}))\over p}\right\rceil+2, \left\lceil{{\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^p_p,{\mathcal H}))+1\over p-1}\right\rceil+1  \right).$$ 
\end{corollary}
\begin{proof}
Since ${\mathcal H}$ has at least one edge, we have $\omega({\mathcal H})\geq p$. 
Therefore, if we set $r=p$ in Theorem~\ref{localhyper}, then the assertion follows immediately.
\end{proof}
Note that if ${\mathcal H}=\KG^p(\F)$ is a nonempty hypergraph, then, in view of Theorem~\ref{inequalities},
we have 
$${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^p_p,{\mathcal H}))+p\geq |V(\F)|-\alt_p(\F).$$
This implies that the previous corollary is an improvement of Meunier's lower bound for the
local chromatic number of $\KG^p(\F)$

\subsection{\bf Plan}
Section~\ref{intro} contains some backgrounds and essential definitions used 
elsewhere in the paper.
In Section~\ref{definition}, we present some new topological 
tools, which help us for the proofs of main results. 
Section~\ref{sec:proofs} is devoted to the proofs of 
Theorem~\ref{maincolorfulindex} and Theorem~\ref{inequalities}.


\section{\bf Preliminaries}\label{intro}

\subsection{\bf Some Topological Indices}

We assume basic knowledge in combinatorial algebraic topology. 
Here, we are going to bring a brief review of some essential notations and definitions, which will be needed throughout the paper. For more, one can see the book written by Matou{\v{s}}ek~\cite{MR1988723}. Also, the definitions of box-complex, Hom-complex, and cross-index
will be generalized to $\Z_p$-box-complex, $\Z_p$-Hom-complex, and $\Z_p$-cross-index, respectively.

Let $\mathbb{G}$ be a finite nontrivial group acting on a topological space $X$.
We call $X$ a {\it topological $\mathbb{G}$-space} if for each $g\in \mathbb{G}$, the 
map  $g:X\longrightarrow X$ with $x\mapsto g\cdot x$ is continuous. A {\it free topological 
$\mathbb{G}$-space $X$} is a topological $\mathbb{G}$-space for which $\mathbb{G}$ 
acts on it freely, i.e., for each $g\in \mathbb{G}\setminus\{\boldsymbol{1}\}$, 
the map $g:X\longrightarrow X$ has no fixed point. 
Here, by $\boldsymbol{1}$,  we mean the identity element of the group 
$\mathbb{G}$.  
For two topological $\mathbb{G}$-spaces $X$ and $Y$,
a map $f:X\longrightarrow Y$ is called a {\it $\mathbb{G}$-equivariant map}, if $f(g\cdot x)=g\cdot f(x)$ for each $g\in \mathbb{G}$ and $x\in X$. Also, any continuous $\mathbb{G}$-map simply is called a {\it $\mathbb{G}$-map}.
 We write $X\stackrel{\mathbb{G}}{\longrightarrow} Y$ to mention that there is a $\mathbb{G}$-map from $X$ to $Y$. 


Simplicial complexes provide
a bridge between combinatorics and topology. A simplicial complex can be viewed
as a combinatorial object, called abstract simplicial complex, or as a topological
space, called geometric simplicial complex. Here, we just remind the definition of an
abstract simplicial complex. However, we assume that the reader is familiar with the concept of how an abstract simplicial complex and its geometric realization are connected to each other. 
A {\it simplicial complex} is a pair $(V,K)$, where $V$ is a finite set and $K$ is a family of subsets of $V$ such that if $F\in K$ and $F'\subseteq F$, then $F'\in K$. 
Any set in $K$ is called a {\it simplex of $K$.} 
Since we may assume that $V=\bigcup\limits_{F\in K}F$,
we can write $K$ instead of $(V, K)$.
The {\it dimension of $K$} is defined as follows:
$${\rm dim}(K)=\max\{|F|-1:\; F\in K\}.$$ 
%The geometric realization of $K$ is denoted by $||K||$.
For two simplicial complexes $C$ and $K$, by a 
{\it simplicial map $f:C\longrightarrow K$,} 
we mean a map from $V(C)$ to $V(K)$ such that the image of any 
simplex of $C$ is a simplex of $K$. 
For a nontrivial finite group $\mathbb{G}$, a {\it simplicial $\mathbb{G}$-complex} $K$ 
is a simplicial complex with a $\mathbb{G}$-action on its vertices such that 
each $g\in \mathbb{G}$ induces a simplicial map from $K$ to $K$, that is
the map which maps $v$ to $g\cdot v$ for each $v\in V(K)$.
If for each $g\in \mathbb{G}\setminus\{\boldsymbol{1}\}$, there is 
no fixed simplex under the simplicial map made by $g$, then $K$ is called a 
{\it free simplicial $\mathbb{G}$-complex.} 
%For a simplicial $\mathbb{G}$-complex $K$, if we take the affine extension, then $K$ is free if and only if $||K||$ is free.
A map $f:C\longrightarrow K$ 
is called {\it $\mathbb{G}$-equivariant}, if $f(g\cdot v)=g\cdot f(v)$ 
for each $g\in \mathbb{G}$ and $v\in V(C)$.
For two simplicial $\mathbb{G}$-complexes $C$ and $K$, a 
{\it simplicial $\mathbb{G}$-map} is a $\mathbb{G}$-equivariant simplicial map form 
$C$ to $K$.  By ${\rm Im(f)}$, we mean a simplicial 
$\mathbb{G}$-subcomplex of K, 
whose vertex set is $f(V(C))$ and whose simplex set is $\left\{\tau\in K\;:\; \exists\;\sigma\in C\;\mbox{ such that } \;f(\sigma)=\tau\right\}.$ 
%We write  $C\stackrel{\mathbb{G}} {\longrightarrow} K$, if there is a simplicial $\mathbb{G}$-map from $C$ to $K$. 


For an integer $n\geq 0$ and a nontrivial finite group $\mathbb{G}$, an
{\it $E_n \mathbb{G}$ space} is a  
free $(n-1)$-connected $n$-dimensional simplicial $\mathbb{G}$-complex. 
A concrete example of an $E_n \mathbb{G}$ space is the $(n+1)$-fold 
join $\mathbb{G}^{*(n+1)}$. 
As a topological space $\mathbb{G}^{*(n+1)}$ is a $(n+1)$-fold join of 
an $(n+1)$-point discrete space.  
This is known that for any two $E_n \mathbb{G}$ space $X$ and $Y$,
there is a $\mathbb{G}$-map from $X$ to $Y$.

For a $\mathbb{G}$-space $X$, define 
$${\rm ind}_{\mathbb{G}}(X)=\min\{n:\; X\stackrel{\mathbb{G}}{\longrightarrow}
E_n\mathbb{G}\}.$$
Note that here $E_n \mathbb{G}$ can be any $E_n \mathbb{G}$ space since there is a 
$\mathbb{G}$-map between any two $E_n \mathbb{G}$ spaces, see~\cite{MR1988723}. Also, for a simplicial 
complex $K$, by ${\rm ind}_{\mathbb{G}}(K)$, we mean 
${\rm ind}_{\mathbb{G}}(||K||)$, where $||K||$ denotes the geometric realization of $K$. 
One must note that any simplicial $\mathbb{G}$-map from $C$ to $K$ induces a  
$\mathbb{G}$-map from $||C||$ to $||K||$. 
Throughout the paper,
for $\mathbb{G}=\mathbb{Z}_2$, we would rather use ${\rm ind}(-)$ 
instead of ${\rm ind}_{\mathbb{Z}_2}(-)$.
In the following, we remind some of the properties of the $\mathbb{G}$-index, which will be used throughout the paper. \\

\noindent{\bf Properties of the $\mathbb{G}$-index.} \cite{MR1988723} Let $\mathbb{G}$ be a finite nontrivial group.
\begin{enumerate}[label={\rm (\roman*)}]
\item\label{P1} ${\rm ind}_{\mathbb{G}}(X)>{\rm ind}_{\mathbb{G}}(Y)$ implies $X\stackrel{\mathbb{G}}{\centernot\longrightarrow} Y$.
\item\label{P2} ${\rm ind}_{\mathbb{G}}(E_n \mathbb{G})=n$ for any $E_n \mathbb{G}$ space.
\item\label{P3} ${\rm ind}_{\mathbb{G}}(X*Y)\leq {\rm ind}_{\mathbb{G}}(X)+{\rm ind}_{\mathbb{G}}(Y)+1$.
%\item\label{P4} If $X$ is $(n-1)$-connected, then ${\rm ind}_{\mathbb{G}}(X)\geq n$.
\item\label{P5} If $K$ is a free simplicial $\mathbb{G}$-complex of dimension $n$, then ${\rm ind}_{\mathbb{G}}(K)\leq n$.
\end{enumerate}
Note that  by the second property, since for each $n$, the simplicial $\mathbb{G}$-complex 
$\mathbb{G}^{*n}$ is an $E_{n-1} \mathbb{G}$ space, we have ${\rm ind}_{\mathbb{G}}(\mathbb{G}^{*n})=n-1$. We will use this fact throughout the paper without any further discussion. 
\subsection{{\bf $\mathbb{Z}_p$-Box-Complex, $\mathbb{Z}_p$-Poset, and $\mathbb{Z}_p$-Hom-Complex}}\label{Boxdefin}
In this subsection, for any $r$-uniform hypergraph ${\mathcal H}$, we shall define 
two objects namely $\mathbb{Z}_p$-box-complex of ${\mathcal H}$ and $\mathbb{Z}_p$-Hom-complex of ${\mathcal H}$,  which the first one is a
simplicial $\mathbb{Z}_p$-complex and the second one is a $\mathbb{Z}_p$-poset.
Moreover, for any $\mathbb{Z}_p$-poset $P$, we assign a combinatorial index to $P$ 
called the $\mathbb{Z}_p$-cross-index of $P$. 
However, as one can see 
in~\cite{MR857448,Iriye20131333,Jonsson2008}, plenty of simplicial complexes have been 
associated to graphs and hypergraphs, used for studying the coloring properties of graphs and hypergraphs. 
Before defining $\mathbb{Z}_p$-box-complex and $\mathbb{Z}_p$-Hom-complex, let us 
review some of these simplicial complexes and briefly describe  their relations with our definitions of 
$\mathbb{Z}_p$-box-complex and $\mathbb{Z}_p$-Hom-complex in this paper. 

For any $r$-uniform hypergraph $\mathcal{H}$, 
Alon, Frankl, and Lov{\'a}sz~\cite{MR857448} defined a kind of 
$\mathbb{Z}_p$-box-complex, denoted $C(\mathcal{H})$, and by using the connectivity of 
this simplicial complex, they presented a 
lower bound for the chromatic number of $\mathcal{H}$ provided that $r$ is odd. 
They used their lower bound for finding the chromatic number of Kneser hypergraphs 
$\KG^r(n,k)$, solving a conjecture posed by Erd{\H{o}}s~\cite{MR0465878}.
Inspired by the works of Lov{\'a}sz~\cite{MR514625} and 
Alon, Frankl, and Lov{\'a}sz~\cite{MR857448}, K{\v{r}}{\'{\i}}{\v{z}}~\cite{MR1081939} introduced a $\mathbb{Z}_r$-poset 
called the resolution of $\mathcal{H}$. He used this $\mathbb{Z}_r$-poset for introducing 
another version of box complexes, which is called ``resolution complex''.  
Using this box complex, he presented some lower bounds for the chromatic number of 
$r$-uniform hypergraphs. 
Although, the definition of this $\mathbb{Z}_r$-poset
is the same as our definition of $\mathbb{Z}_r$-Hom-complex of ${\mathcal H}$,  
we will use this $\mathbb{Z}_r$-poset for a completely different purpose from  
what K{\v{r}}{\'{\i}}{\v{z}} did in his paper.  

Moreover, for a graph $G$, there is another famous simplicial complex 
${\rm Hom}(K_2,G)$, which is introduced by Lov{\'a}sz~\cite{MR514625}.  
Iriye and Kishimoto~\cite{Iriye20131333} extended Lov{\'a}sz's definition to $r$-uniform 
hypergraphs. For any $r$-uniform hypergraph $\mathcal{H}$, they introduced a simplicial 
complex ${\rm Hom}(K_r^{(r)},\mathcal{H})$, and used it for providing some lower bounds for the
chromatic number of $r$-uniform hypergraphs provided that $r$ is prime. One must note 
that Lov{\'a}sz's definition and its extension by Iriye and Kishimoto  are 
completely  different from the 
definition of $\mathbb{Z}_r$-Hom-complex ${\rm Hom}(K_r^r,\mathcal{H})$ in this paper. 
%Actually, ${\rm Hom}(K_r^{(r)},\mathcal{H})$ defined by Iriye and Kishimoto is a simplicial complex while our definition of Hom-complex results in a $\mathbb{Z}_p$-posets.  
Also, Thansri~\cite{Thansri2012} compared the homotopy type of the simplicial complex ${\rm Hom}(K_r^{(r)},\mathcal{H})$ defined by Iriye and Kishimoto and the box-complex $C(\mathcal{H})$ defined by Alon, Frankl, and Lov{\'a}sz~\cite{MR857448}.  \\ 



\noindent{\bf $\mathbb{Z}_p$-Box-Complex.}
Let $r$ be a positive integer and $p$ be a prime number, where $2\leq r\leq p$.
For an $r$-uniform hypergraph ${\mathcal H}$, define the {\it $\mathbb{Z}_p$-box-complex of ${\mathcal H}$,} denoted ${\rm B}_0({\mathcal H},{\mathbb{Z}_p})$,  
to be a simplicial complex with the vertex set $\biguplus\limits_{i=1}^pV({\mathcal H})=\mathbb{Z}_p\times V({\mathcal H})$ and  the simplex set consisting of all
$\{\omega^1\}\times U_1\cup\cdots\cup \{\omega^p\}\times U_p$,
where 
 \begin{itemize}
\item $U_1,\ldots,U_p$ are pairwise disjoint subsets of $V({\mathcal H})$ and
%\item $\displaystyle\bigcup_{i=1}^p U_i\neq\varnothing$, and
\item the hypergraph ${\mathcal H}[U_1,U_2,\ldots,U_p]$
is a complete $r$-uniform $p$-partite hypergraph.
 \end{itemize}
Note that some of the sets $U_i$'s might be empty. 
In fact, if $U_1,\ldots,U_p$ are pairwise disjoint subsets of $V({\mathcal H})$ and the number of nonempty $U_i$'s is less than $r$, then ${\mathcal H}[U_1,U_2,\ldots,U_p]$ is a complete $r$-uniform $p$-partite hypergraph and thus
$\{\omega^1\}\times U_1\cup\cdots\cup \{\omega^p\}\times U_p\in {\rm B}_0({\mathcal H},{\mathbb{Z}_p})$. For each $\varepsilon\in{\mathbb{Z}_p}$ 
and each $(\varepsilon',v)\in V({\rm B}_0(\mathcal{H},{\mathbb{Z}_p}))$, define  $\varepsilon\cdot(\varepsilon',v)=(\varepsilon\cdot\varepsilon',v)$. One can see that this action makes   
${\rm B}_0({\mathcal H},{\mathbb{Z}_p})$ a free simplicial $\mathbb{Z}_p$-complex. 
Whenever $\mathcal{H}=G$ is a graph ($2$-uniform hypergraph), 
the $\Z_2$-box-complex ${\rm B}_0(G,\Z_2)$ is 
extensively 
studied in the literature, where it is known as the box complex of $G$, denoted ${\rm B_0}(G)$, 
for instance, see~\cite{MR2279672,MR2351519}.  
This simplicial complex is used for introducing some lower bounds for the chromatic 
number of graphs, see~\cite{MR2279672}. \\

\noindent{\bf $\mathbb{Z}_p$-Poset.}
A partially ordered set, or simply a {\it poset}, is defined as an ordered pair $P=(V(P), \preceq)$, where  $V(P)$ is a nonempty set called the ground set of $P$ and $\preceq$ is a partial order on $V(P)$.
For two posets $P$ and $Q$, 
by an order-preserving map $\phi:P\longrightarrow Q$, we mean  a map $\phi$  
from  $V(P)$ to $V(Q)$  such that for each $u,v\in V(P)$, if $u\preceq v$, then $\phi(u)\preceq \phi(v)$.
 A poset $P$ is called a {\it  $\mathbb{Z}_p$-poset}, if $\mathbb{Z}_p$ acts on $V(P)$ 
 and furthermore,
for each $\varepsilon\in \mathbb{Z}_p$, the map $\varepsilon:V(P)\longrightarrow V(P)$ with $v\mapsto \varepsilon\cdot v$ is an automorphism of $P$ (order-preserving bijective map).
If for each $\varepsilon\in \mathbb{Z}_p\setminus\{\boldsymbol{1}\}$, this map has no fixed  point, then $P$ is called a {\it  free $\mathbb{Z}_p$-poset}. 
Here, by $1$, we mean the identity element of $\mathbb{Z}_p$, i.e., $1=\omega^0$. 
For two $\mathbb{Z}_p$-posets $P$ and $Q$, 
by an order-preserving $\mathbb{Z}_p$-map $\phi:P\longrightarrow Q$, we mean 
 an order-preserving map from  $V(P)$ to $V(Q)$ such that for each $v\in V(P)$ and $\varepsilon\in \mathbb{Z}_p$, we have $\phi(\varepsilon\cdot v)=\varepsilon\cdot\phi(v)$.

For a nonnegative integer $n$ and a prime number $p$, let $Q_{n,p}$ be a free
$\mathbb{Z}_p$-poset with the ground set $\mathbb{Z}_p\times[n+1]$ such that for each  
two elements $(\varepsilon,i),(\varepsilon',j)\in Q_{n,p}$,  we have 
$(\varepsilon,i)<_{Q_{n,p}}(\varepsilon',j)$ whenever 
$i<j$. Clearly, $Q_{n,p}$ is a   free $\mathbb{Z}_p$-poset with the action $\varepsilon\cdot(\varepsilon',j)=(\varepsilon\cdot\varepsilon',j)$ for each $\varepsilon\in\mathbb{Z}_p$ and $(\varepsilon',j)\in Q_{n,p}$.
For a $\mathbb{Z}_p$-poset $P$, the {\it $\mathbb{Z}_p$-cross-index of $P$,} denoted
${\rm Xind}_{\mathbb{Z}_p}(P)$, is the least integer $n$ such that there is a 
$\mathbb{Z}_p$-map from $P$ to $Q_{n,p}$. Throughout the paper, for $p=2$,  
we speak about ${\rm Xind}(-)$ rather than ${\rm Xind}_{\mathbb{Z}_2}(-)$. 
It should be mentioned that ${\rm Xind}(-)$ is first defined in~\cite{SiTaZs13}.

Let $P$ be a poset. We can define an {\it order-complex} $\Delta P$ with the vertex set 
same as the ground set of $P$ and simplex set consisting of all chains in $P$. 
One can see that if $P$ is a free $\mathbb{Z}_p$-poset, then $\Delta P$ is a free 
simplicial $\mathbb{Z}_p$-complex. 
Moreover, any order-preserving $\Z_p$-map $\phi:P\longrightarrow Q$ can be lifted 
to a simplicial 
$\mathbb{Z}_p$-map from $\Delta P$ to $\Delta Q$. Clearly, there is a simplicial 
$\mathbb{Z}_p$-map from $\Delta Q_{n,p}$ to $\mathbb{Z}_p^{*(n+1)}$ (identity map). 
Therefore,
if ${\rm Xind}_{\mathbb{Z}_p}(P)=n$, then
we have a simplicial $\mathbb{Z}_p$-map from $\Delta P$ to $\mathbb{Z}_p^{*(n+1)}$. This 
implies that 
${\rm Xind}_{\mathbb{Z}_p}(P)\geq {\rm ind}_{\mathbb{Z}_p}(\Delta P)$. 
%{\color{red} Throughout the paper, for each $(\varepsilon, j)\in Q_{n,p}$, when we speak about the sign of $(\varepsilon, j)$ and the absolute value of  $(\varepsilon, j)$, we mean $\varepsilon$ and $j$, respectively.} 

\begin{theorem}{\rm \cite{AliHajiMeu2016}}\label{altercrossindex}
Let $P$ be a free ${\mathbb Z}_2$-poset and $\phi:P\longrightarrow Q_{s,2}$ be an 
order-preserving ${\mathbb Z}_2$-map. Then $P$ contains a chain $p_1\prec_P\cdots\prec_Pp_{ k}$ such that $k= {\rm Xind}(P)+1$ and
the signs of $\phi(p_i)$ and $\phi(p_{i+1})$ differ 
for each $i\in[k-1]$. Moreover, if $s= {\rm Xind}(P)$, then for any $(s+1)$-tuple 
$(\varepsilon_1,\ldots,\varepsilon_{s+1})\in\mathbb{Z}_2^{s+1}$, there is at least one chain 
$p_1\prec_P\cdots\prec_Pp_{ s+1}$ such that $\phi(p_i)=(\varepsilon_i,i)$ for each $i\in[s+1]$.
\end{theorem}

\noindent{\bf $\mathbb{Z}_p$-Hom-Complex.}
Let ${\mathcal H}$ be an $r$-uniform hypergraph. Also, let $p$ be a prime number
such that $2\leq r\leq p\leq\omega(\mathcal{H})$. 
The {\it $\mathbb{Z}_p$-Hom-complex} ${\rm Hom}(K^r_p,{\mathcal H})$ is a free 
$\mathbb{Z}_p$-poset with the ground set consisting of all ordered $p$-tuples 
$(U_1,\ldots,U_p)$, where $U_i$'s are nonempty pairwise disjoint subsets of $V$ and 
${\mathcal H}[U_1,\ldots,U_p]$ is a complete $r$-uniform $p$-partite hypergraph. 
For two $p$-tuples $(U_1,\ldots,U_p)$ and $(U'_1,\ldots,U'_p)$ in 
${\rm Hom}(K^r_p,{\mathcal H})$, we define $(U_1,\ldots,U_p)\preceq(U'_1,\ldots,U'_p)$ 
if $U_i\subseteq U'_i$ for each $i\in[p]$. 
Also, for each $\omega^i\in \mathbb{Z}_p=\{\omega^1,\ldots,\omega^p\}$, let 
$\omega^i\cdot (U_1,\ldots,U_p)=(U_{1+i},\ldots,U_{p+i})$, where $U_j=U_{j-p}$ for $j>p$. 
Clearly, this action is a free $\mathbb{Z}_p$-action on ${\rm Hom}(K^r_p,{\mathcal H})$. 
Consequently, ${\rm Hom}(K^r_p,{\mathcal H})$ is a free $\mathbb{Z}_p$-poset with 
this $\mathbb{Z}_p$-action. Note that since $p\leq \omega ({\mathcal H})$, the ground set of 
${\rm Hom}(K^r_p,{\mathcal H})$ is not empty.  

For a nonempty graph $G$ and $p=2$, we would rather use ${\rm Hom}(K_2,G)$
instead of ${\rm Hom}(K^2_2,G)$. Also, it should be mentioned that ${\rm Hom}(K_2,G)$ is first defined in~\cite{SiTaZs13}, known as the Hom-complex of $G$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{\bf Notations and Tools}\label{definition}

For a simplicial complex $K$, by $\sd K$, we mean {\it the first barycentric subdivision of $K$}.
It is the order-complex obtained from the poset consisting of all nonempty simplices in $K$
ordered by inclusion. 
%It is the simplicial complex, whose vertex set is the set of nonempty simplices of $K$ and whose simplices are the collections of simplices of $K$ which are pairwise comparable by inclusion.  
Throughout the paper, by $\sigma_{t-1}^{r-1}$, we mean the 
$(t-1)$-dimensional simplicial complex with vertex set $\mathbb{Z}_r$,   
containing all $t$-subsets 
of $\mathbb{Z}_r$ as its maximal simplices. 
The join of two simplicial complexes $C$ and $K$, denoted $C*K$, is a simplicial complex with the vertex set $V(C)\uplus V(K)$ and such that the set of its simplices is
$\{F_1\biguplus F_2:\; F_1\in C\mbox{ and } F_2\in K\}$.
Clearly, we can see $\mathbb{Z}_r$ as a $0$-dimensional simplicial complex. 
Note that the vertex set of simplicial complex $\sd\mathbb{Z}_r^{*\alpha}$ can 
be identified with $(\mathbb{Z}_r\cup\{0\})^\alpha\setminus\{\zero\}$ and the vertex 
set of $(\sigma^{r-1}_{t-1})^{*n}$ is the set of all pairs $(\varepsilon,i)$, where 
$\varepsilon\in \mathbb{Z}_r$ and $i\in [n]$.
Also, for each simplex $\tau\in (\sigma^{p-1}_{p-2})^{*m}$ and 
for each $\varepsilon\in \mathbb{Z}_p$, define
$\tau^\varepsilon=\left\{(\varepsilon,j):\; (\varepsilon,j)\in \tau\right\}.$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{{\bf $\mathbb{Z}_p$-Tucker-Ky Fan lemma}}
The famous Borsuk--Ulam~theorem has many interesting generalizations, which have 
been used extensively for investigating  graphs coloring properties. For examples, 
Tucker's lemma~\cite{MR0020254}, the $Z_p$-Tucker Lemma~\cite{MR1893009}, and 
Ky~Fan's lemma~\cite{MR0051506} are some of these interesting generalizations. 
%Some of these interesting generalizations are Tucker's~lemma~\cite{MR0020254}, $Z_p$-Tucker Lemma~\cite{MR1893009}, and Tucker-Ky Fan~\cite{MR0051506}.  
For more details about the Borsuk--Ulam theorem and its generalizations, we refer the reader to \cite{MR1988723}.

Indeed, Tucker's~lemma is a combinatorial counterpart of the Borsuk--Ulam theorem. There are several interesting and surprising applications of Tucker's~lemma in combinatorics, including a combinatorial proof of the Lov{\'a}sz--Kneser 
theorem by Matou{\v{s}}ek \cite{MR2057690}. 
\begin{lemma}\label{tuckeroriginal}
{\rm(Tucker's~lemma \cite{MR0020254}).} 
Let $m$ and $n$ be positive integers and
$\lambda:\{-1,0,+1\}^n\setminus \{\zero\} \longrightarrow \{\pm 1, \pm 2,\ldots,\pm m\}$
be a map satisfying the following properties:
\begin{itemize}
\item for any $X\in \{-1,0,+1\}^n\setminus \{\zero\}$, we have $\lambda(-X)=-\lambda(X)$ {\rm (}a
      $Z_2$-equivariant map{\rm ),}
\item no two signed vectors $X$ and $Y$ are such that
$X\subseteq Y$ and $\lambda(X) =-\lambda(Y)$.
\end{itemize}
Then, we have $m \geq n$.
\end{lemma}
Another interesting generalization of the Borsuk--Ulam Theorem is the well-known  
Ky~Fan's lemma~\cite{MR0051506}. This generalization ensures that with the 
same hypotheses as in Lemma~\ref{tuckeroriginal}, there are odd number of chains
$X_1\subseteq X_2\subseteq \cdots \subseteq X_n$ such that $$\{\lambda(X_1),\ldots,\lambda(X_n)\}=\{+c_1,-c_2,\ldots,
(- 1)^{n-1}c_n\},$$ where  $1\leq c_1 < \cdots < c_n \leq m$.
Ky~Fan's lemma has been used in several articles to
study some coloring properties of graphs, see \cite{AliHajiMeu2016,MR2763055,MR2837625}.
There are also some other generalizations of
Tucker's~lemma. 
A $\mathbb{Z}_p$ version of Tucker's~lemma, called the $\mathbb{Z}_p$-Tucker Lemma, is proved by Ziegler~\cite{MR1893009} and extended by Meunier~\cite{MR2793613}.
In the next subsection, we present a $\mathbb{Z}_p$ version of Ky~Fan's lemma, 
which is called the $\mathbb{Z}_p$-Tucker-Ky Fan Lemma.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{{\bf New Generalizations of Tucker's~Lemma}} 
This subsection is devoted to introduce some new topological tools, which will be 
used elsewhere  in the paper. 
Note that 
if we set $\mathbb{Z}_2=\{-1,+1\}$, then 
any map $\lambda$ satisfying the conditions of Tucker's~lemma (Lemma~\ref{tuckeroriginal})
can be considered as a $\mathbb{Z}_2$-simplicial map from $\sd \mathbb{Z}_2^{*n}$ to 
$\mathbb{Z}_2^{*m}$. 
In this point of view, Ky~Fan's~lemma says that for any such a 
map $\lambda$, there is at least one $(n-1)$-dimensional simplex 
$\sigma\in (\sd \mathbb{Z}_2^{*n})$ such that  
$\lambda(\sigma)=\{+c_1,-c_2,\ldots,
(- 1)^{n-1}c_n\}$,  where  $1\leq c_1 < \cdots < c_n \leq m$. 
Note that $ \mathbb{Z}_2^{*m}$ is a sub-complex of $(\sigma^{p-1}_{p-2})^{*m}$ and ${\rm ind}_{\mathbb{Z}_2}(\sd \mathbb{Z}_2^{*n})=n-1$. Therefore, the 
next lemma can be considered as a $\mathbb{Z}_p$-generalization of Ky~Fan's~lemma. 
\begin{lemma}\label{genfanlemma}
Let $C$ be a free  simplicial $\mathbb{Z}_p$-complex 
such that ${\rm ind}_{\mathbb{Z}_p}(C)\geq t$
and let $\lambda:C\longrightarrow (\sigma^{p-1}_{p-2})^{*m}$ be a simplicial $\mathbb{Z}_p$-map.
Then there is at least one $t$-dimensional simplex $\sigma\in C$ such that $\tau=\lambda(\sigma)$ is a $t$-dimensional simplex and for each $\varepsilon\in \mathbb{Z}_p$, we have 
$\lfloor{t+1\over p}\rfloor\leq |\tau^\varepsilon|\leq\lceil{t+1\over p}\rceil.$ 
\end{lemma}

Before starting the proof of Lemma~\ref{genfanlemma}, 
we need to introduce three functions. These functions are crucial for the rest of  the paper  
and we will use them throughout the paper in several times. 
Let $m$ be a positive integer and $p$ be a prime number.
We have already noted that $(\sigma^{p-1}_{p-2})^{*m}$ is a free simplicial 
$\mathbb{Z}_p$-complex with the vertex set $\mathbb{Z}_p\times [m]$.

\noindent{\bf The value function $l(-)$.}
Let $\tau\in (\sigma^{p-1}_{p-2})^{*m}$ be a simplex. For each $\varepsilon\in \mathbb{Z}_p$, we remind the reader that 
$\tau^\varepsilon=\left\{(\varepsilon,j):\; (\varepsilon,j)\in \tau\right\}.$ 
Now,  define 
$$l(\tau)=\max\left\{\displaystyle|\bigcup_{\varepsilon\in\mathbb{Z}_p} B^\varepsilon|:\; \forall\varepsilon\in\mathbb{Z}_p,\; B^\varepsilon\subseteq \tau^\varepsilon\mbox{ and } \forall \varepsilon_1,\varepsilon_2\in\mathbb{Z}_p,\; |\;|B^{\varepsilon_1}|-|B^{\varepsilon_2}|\;|\leq 1 \right\}.$$
Note that if we set $h(\tau)=\ds\min_{\varepsilon\in \mathbb{Z}_p}|\tau^\varepsilon|$, then
$$l(\tau)=p\cdot h(\tau)+|\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|>h(\tau)\}|.$$
It is clear that the function $l(-)$ is monotone, i.e.,  if $\tau_1\subseteq \tau_2$, 
then $l(\tau_1)\leq l(\tau_2)$.  
Also, the following remark can be readily obtained from the definition of the function $l(-)$. 
\begin{remark}\label{remark}
If there is a simplex $\tau_1\in (\sigma^{p-1}_{p-2})^{*m}$ such that 
$l(\tau_1)\geq l$, then there is a simplex $\tau\subseteq \tau_1$ with 
$|\tau|=l(\tau)=l$ and such that for each 
$\varepsilon\in\mathbb{Z}_p$, we have 
$\lfloor{l\over p}\rfloor\leq |\tau^\varepsilon|\leq\lceil{l\over p}\rceil$.
\end{remark} 


\noindent{\bf The sign functions $s(-)$ and $s_0(-)$.} 
For an $a\in[m]$, 
let $W_a$ be the set of all nonempty simplices $\tau\in (\sigma_{p-2}^{p-1})^{*m}$ such that $|\tau^\varepsilon|\in\{0,a\}$ for each $\varepsilon\in\mathbb{Z}_p$. Let $W=\displaystyle\bigcup_{a=1}^{m}W_a$. 
Choose an arbitrary $\mathbb{Z}_p$-equivariant map $s:W\longrightarrow \mathbb{Z}_p$. 
Also, consider a $\mathbb{Z}_p$-equivariant map $s_0:\sigma_{p-2}^{p-1}\longrightarrow \mathbb{Z}_p$.
Note that since $\mathbb{Z}_p$ acts freely on both $W$ and $\sigma_{p-2}^{p-1}$, 
these maps can be easily built by choosing one representative in each orbit. We should emphasize that both functions $s(-)$ and $s_0(-)$ are first introduced in~\cite{Meunier14}.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of Lemma~{\rm\ref{genfanlemma}}]
For simplicity of notation, let $K={\rm Im}(\lambda)$. 
In view of Remark~\ref{remark}, to prove the assertion, it is enough to show that there 
is a $t$-dimensional simplex $\tau\in K$ such that $l(\tau)\geq t+1$.  
For a contradiction, suppose that there is no such a $t$-dimensional simplex.
Therefore, for each simplex $\tau$ of $K$, we have $l(\tau)\leq t$. 

Let $\Gamma:\sd K\longrightarrow \mathbb{Z}_p^{*t}$  be a map which will be defined 
as follows. Let $\tau$ be a vertex of $\sd K$.  
Consider two following cases depending on the value of 
$h(\tau)=\ds\min_{\varepsilon\in \mathbb{Z}_p}|\tau^\varepsilon|$. 
\begin{enumerate}[label={\rm (\roman*)}] 
\item   If $h(\tau)=0$, then define $\bar{\tau}=\{\varepsilon\in \mathbb{Z}_p:\; 
                      \tau^\varepsilon=   \varnothing\}\in \sigma^{p-1}_{p-2}$ and 
                      $$\Gamma(\tau)=\left(s_0(\bar\tau), l(\tau)\right).$$
              
\item  If $h(\tau)> 0$, then define $\bar{\tau}=\displaystyle
	        \bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h(\tau)\}} \tau^\varepsilon\in W$ 
	        and $$\Gamma(\tau)=\left(s(\bar\tau), l(\tau)\right).$$
\end{enumerate}

First, we show that  $\Gamma$ is a simplicial $\mathbb{Z}_p$-map from $\sd K$ to 
$\mathbb{Z}_p^{*t}$. Since both functions $s(-)$ and $s_0(-)$ are $\mathbb{Z}_p$-equivariant, it is clear that  $\Gamma$ is a $\mathbb{Z}_p$-equivariant map. 
For a contradiction, suppose that  $\Gamma$ is~not a simplicial map. Therefore, 
there are $\tau,\tau' \in \sd K$
such that $\tau\subsetneq\tau'$, $\Gamma(\tau)=(\varepsilon_1,\beta)$, and 
$\Gamma(\tau')=(\varepsilon_2,\beta)$, where $\varepsilon_1\neq \varepsilon_2$. 
Clearly, in view of the definition of $\Gamma$, we have $l(\tau)=l(\tau')=\beta$.
Now, we consider three different cases.
\begin{enumerate}[label={\rm (\roman*)}]
\item If $h(\tau)=h(\tau')=0$, then since $\tau\subsetneq\tau'$ and $$\varepsilon_1=s_0(\{\varepsilon\in \mathbb{Z}_p:\; 
         \tau^\varepsilon=   \varnothing\})\neq s_0(\{\varepsilon\in \mathbb{Z}_p:\;  
         {\tau'}^\varepsilon=   \varnothing\})=\varepsilon_2,$$ 
          we have    $ \{\varepsilon\in \mathbb{Z}_p:\;  
         {\tau'}^\varepsilon=   \varnothing\}\subsetneq\{\varepsilon\in \mathbb{Z}_p:\; 
         \tau^\varepsilon=   \varnothing\}$. This implies that 
         $$l(\tau')=p-|\{\varepsilon\in \mathbb{Z}_p:\; 
         {\tau'}^\varepsilon=   \varnothing\}|>p-|\{\varepsilon\in \mathbb{Z}_p:\; 
         \tau^\varepsilon=   \varnothing\}|=l(\tau),$$ 
         a contradiction.\\

\item If $h(\tau)=0$ and $h(\tau')>0$, then  
	$l(\tau)\leq p-1$ and $l(\tau')\geq p$, contradicting $l(\tau)=l(\tau')$.\\

\item If $h(\tau)>0$ and $h(\tau')>0,$ then 
	 %Note that 
	 $l(\tau)=p\cdot h(\tau)+|\{\varepsilon\in\mathbb{Z}_p: |\tau^\varepsilon|>h(\tau)\}|$ and 
	 $ l(\tau')=p\cdot h(\tau')+|\{\varepsilon\in\mathbb{Z}_p: 
	 |{\tau'}^\varepsilon|>h(\tau')\}|.$
For this case, two different sub-cases will be distinguished.
	 \begin{itemize}
	 \item[(a)] $h(\tau)=h(\tau')=h$. Since  
	 $$\varepsilon_1=s(\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h\}} \tau^\varepsilon)\neq s(\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\}} {\tau'}^\varepsilon)=\varepsilon_2,$$
	 we must have  $$\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h\}} \tau^\varepsilon\neq \displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\}} {\tau'}^\varepsilon.$$
	 Note that $\tau\subseteq \tau'$ and $\min\limits_{\varepsilon\in \mathbb{Z}_p}|\tau^\varepsilon|=\min\limits_{\varepsilon\in \mathbb{Z}_p}|{\tau'}^\varepsilon|.$ Therefore, we should have
	 $$
	 \{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\} \subsetneq \{\varepsilon\in\mathbb{Z}_p:\; |{\tau}^\varepsilon|=h\}$$
	 and consequently $l(\tau)<l(\tau')$, which is~not possible.
	 \item[(b)] $h(\tau)<h(\tau')$. Then, one can see that  
	 $$l(\tau)\leq p\cdot h(\tau)+p-1< p\cdot (h(\tau)+1)\leq l(\tau'),$$
	 which is a contradiction.	
	 \end{itemize}
\end{enumerate}
Therefore, $\Gamma$ is a simplicial $\mathbb{Z}_p$-map from $\sd K$ to $\mathbb{Z}_p^{*t}$. 
On the other hand, $\lambda$ can naturally be lifted to a simplicial $\mathbb{Z}_p$-map $\bar\lambda:\sd C\longrightarrow \sd K$. 
Thus $\Gamma\circ\bar\lambda$ is a  simplicial $\mathbb{Z}_p$-map from $\sd C$ to
$\mathbb{Z}_p^{*t}$. 
In view of Property~\ref{P1} in Properties of the $\mathbb{G}$-index,
it implies that ${\rm ind}_{\mathbb{Z}_p}(C)={\rm ind}_{\mathbb{Z}_p}(\sd C)\leq t-1$, which is~not possible.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The $\mathbb{Z}_p$-Tucker lemma~\cite{MR2793613,MR1893009} is a famous generalization of Tucker's~lemma, having many
applications in Kneser hypergraph coloring, for instance see~\cite{AliHajiMeu2016,2013arXiv1302.5394A,MR2971704,MR2763055,MR2837625,MR2057690}.  
Although Lemma~\ref{genfanlemma} can be considered as a $\mathbb{Z}_p$ version of Ky~Fan's~lemma, it is~not stated in the simple form as Ky~Fan's~lemma did, which makes  Lemma~\ref{genfanlemma} difficult to use for non-familiars with algebraic topology. 
In the next result, we present a generalization of the $\mathbb{Z}_p$-Tucker lemma, called $\mathbb{Z}_p$-Tucker-Ky Fan lemma, in a form of combinatorial language. 
As an application of this result, we give a simple proof of Meunier's theorem (Theorem~\ref{colorfulhyper}). 
Even though, the only contribution of the $\mathbb{Z}_p$-Tucker-Ky Fan lemma in this paper is to simplify the original proof of Meunier's theorem, 
this lemma is interesting in its own right since it simultaneously generalizes  
Tucker's~lemma, Ky~Fan's~lemma, and the $\mathbb{Z}_p$-Tucker~Lemma. 

\begin{lemma}{\rm ($\mathbb{Z}_p$-Tucker-Ky Fan lemma).}\label{Z_pfanlemma}
Let $m,n,p$ and $\alpha$ be nonnegative integers, where
$m,n\geq 1$, $m\geq \alpha\geq 0$, and $p$ is prime. Let
$$
\begin{array}{crcl}
\lambda: & (\mathbb{Z}_p\cup\{0\})^n\setminus\{\zero\} &\longrightarrow & \mathbb{Z}_p\times[m]\\
& X &\longmapsto & (\lambda_1(X),\lambda_2(X))
\end{array}$$ be a $\mathbb{Z}_p$-equivariant map
satisfying the following conditions. 
\begin{itemize}
\item For $X_1\subseteq X_2\in \left(\mathbb{Z}_p\cup\{0\}\right)^n\setminus\{\zero\}$,
if $\lambda_2(X_1)=\lambda_2(X_2)\leq \alpha$, then $\lambda_1(X_1)=\lambda_1(X_2)$.
\item For $X_1\subseteq X_2\subseteq\cdots \subseteq X_p\in \left(\mathbb{Z}_p\cup\{0\}\right)^n\setminus\{\zero\}$,
if $\lambda_2(X_1)=\lambda_2(X_2)=\cdots=\lambda_2(X_p)\geq\alpha+1$, then 
$$\left|\left\{\lambda_1(X_1),\lambda_1(X_2),\ldots,\lambda_1(X_p)\right\}\right|<p.$$
\end{itemize}
Then there is a chain $$Z_1\subset Z_2\subset\cdots\subset Z_{n-\alpha}\in \left(\mathbb{Z}_p\cup\{0\}\right)^n\setminus\{\zero\}$$
such that 
\begin{enumerate}
\item for each $i\in [n-\alpha]$, $\lambda_2(Z_i)\geq \alpha+1$,
\item  for each $i\neq j\in [n-\alpha]$, $\lambda(Z_i)\neq \lambda(Z_j)$, and
\item\label{condition3} for  each 
$\varepsilon\in\mathbb{Z}_p$, 
$$\left\lfloor{n-\alpha\over p}\right\rfloor\leq \left|\left\{j:\; \lambda_1(Z_j)=\varepsilon\right\}\right|\leq \left\lceil{n-\alpha\over p}\right\rceil.$$
\end{enumerate}
In particular, $n-\alpha\leq (p-1)(m-\alpha)$.
\end{lemma}
\begin{proof}
Clearly, the map $\lambda$ can be considered as a simplicial $\mathbb{Z}_p$-map from $\sd \mathbb{Z}_p^{*n}$ to 
$(\mathbb{Z}_p^{*\alpha})*((\sigma_{p-2}^{p-1})^{*(m-\alpha)}).$
Let $K={\rm Im}(\lambda)$. 
Note that each simplex in $K$ can be represented in a unique form $\sigma\cup\tau$ such that
$\sigma\in \mathbb{Z}_p^{*\alpha}$ and $\tau \in (\sigma_{p-2}^{p-1})^{*m-\alpha}.$
In view of Remark~\ref{remark},  
%and the properties which $\lambda$ satisfies \in, 
to prove the assertion, it suffices to show that there is a simplex $\sigma\cup\tau\in K$ such that  $l(\tau)\geq n-\alpha$. For a contradiction, suppose that for each $\sigma\cup\tau\in K$, we have $l(\tau)\leq n-\alpha-1$.

Define the map 
$\Gamma: \sd K\longrightarrow \mathbb{Z}_p^{*(n-1)}$
such that for each vertex $\sigma\cup\tau\in V(\sd K)$, 
$\Gamma(\sigma\cup\tau)$ is defined as follows.
\begin{itemize}
\item If $\tau=\varnothing$, then $\Gamma(\sigma\cup\tau)=(\varepsilon, j)$, 
where $j$ is the maximum possible value for which $(\varepsilon, j)\in\sigma$.
Note that since $\sigma\in \mathbb{Z}_p^{*\alpha}$,  
there is only one $\varepsilon\in\mathbb{Z}_p$ for which 
 the maximum is attained. Therefore, in this case, the function $\Gamma$ is well-defined.

\item If $\tau\neq\varnothing$. Define $h(\tau)=\ds\min_{\varepsilon\in \mathbb{Z}_p}|\tau^\varepsilon|.$
\begin{enumerate}[label={\rm (\roman*)}]
\item   If $h(\tau)=0$, then define $\bar{\tau}=\{\varepsilon\in \mathbb{Z}_p:\; 
                      \tau^\varepsilon=   \varnothing\}\in \sigma^{p-1}_{p-2}$ and 
                      $$\Gamma(\sigma\cup\tau)=\left(s_0(\bar\tau), \alpha+l(\tau)\right).$$
              
\item  If $h(\tau)> 0$, then define $\bar{\tau}=\displaystyle
	        \bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h(\tau)\}} \tau^\varepsilon\in W$ 
	        and $$\Gamma(\sigma\cup\tau)=\left(s(\bar\tau), \alpha+l(\tau)\right).$$

\end{enumerate}
\end{itemize}
We first show that  $\Gamma$ is a simplicial $\mathbb{Z}_p$-map from $\sd K$ to $\mathbb{Z}_p^{*(n-1)}$. It is clear that  $\Gamma$ is a $\mathbb{Z}_p$-equivariant map.
For a contradiction, suppose that there are $\sigma\cup\tau,\sigma'\cup\tau' \in \sd K$
such that $\sigma\subseteq \sigma'$, $\tau\subseteq\tau'$, $\Gamma(\sigma\cup\tau)=(\varepsilon_1,\beta)$, and $\Gamma(\sigma'\cup\tau')=(\varepsilon_2,\beta)$, where $\varepsilon_1\neq \varepsilon_2$. 
First note that in view of the definition of $\Gamma$ and the assumption $\Gamma(\sigma\cup\tau)=(\varepsilon_1,\beta)$ and $\Gamma(\sigma'\cup\tau')=(\varepsilon_2,\beta)$, the case  $\tau=\varnothing$ and $\tau'\neq\varnothing$ is not possible.
If $\tau'=\varnothing$, then 
$\tau=\tau'=\varnothing$ and we should have 
$(\varepsilon_1,\beta),(\varepsilon_2,\beta)\in\sigma'\in \mathbb{Z}_p^{*\alpha}$, which implies that $\varepsilon_1=\varepsilon_2$, a contradiction.
If $\varnothing\neq \tau\subseteq \tau'$, then in view of definition of $\Gamma$, we should have $l(\tau)=l(\tau')=\beta-\alpha$. Now, similar to the proof of Lemma~\ref{genfanlemma}, we can consider three different cases, each of them resulting in a contradiction. 
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%

Therefore, $\Gamma$ is a simplicial $\mathbb{Z}_p$-map from $\sd K$ to $\mathbb{Z}_p^{*(n-1)}$. Naturally, $\lambda$ can be lifted to a simplicial $\mathbb{Z}_p$-map $\bar\lambda:\sd^2 \mathbb{Z}_p^{*n}\longrightarrow \sd K$. 
Thus $\Gamma\circ\bar\lambda$ is a  simplicial $\mathbb{Z}_p$-map from $\sd^2 \mathbb{Z}_p^{*n}$ to
$\mathbb{Z}_p^{*(n-1)}$.
Therefore, by Property~\ref{P1} in Properties of the $\mathbb{G}$-index, we must have 
$$n-1={\rm ind}_{\mathbb{Z}_p}(\sd^2\mathbb{Z}_p^{*n})\leq 
{\rm ind}_{\mathbb{Z}_p}(\mathbb{Z}_p^{*(n-1)})=n-2,$$
which is~not possible. 
%{\color{red} In view of Dold's theorem~\cite{MR711043,MR1988723},  the dimension of $\mathbb{Z}_p^{*(n-1)}$ should be strictly larger than the connectivity of $\sd^2 \mathbb{Z}_p^{*n}$, that is $n-2>n-2$,which is not possible.}
\end{proof}

As an application of Lemma~\ref{Z_pfanlemma}, we present a short simple proof of 
Meunier's colorful result. 
\begin{theorem}{\rm (Meunier's theorem~\cite{Meunier14})}\label{colorfulhyper}
Let ${\mathcal H}$ be a hypergraph and let $p$ be a prime number. Then any proper coloring $c:V({\rm KG}^p({\mathcal H}))\longrightarrow [L]$ {\rm(}$L$ arbitrary{\rm)} must contain a colorful balanced complete $p$-uniform $p$-partite hypergraph with 
$|V({\mathcal H})|-{\rm alt}_p({\mathcal H})$ vertices.
\end{theorem}
\begin{proof}
Consider a bijection $\pi:[n]\longrightarrow V({\mathcal H})$ such that
${\rm alt}_p({\mathcal H},\pi)={\rm alt}_p({\mathcal H}).$
We are going to define a map $$\begin{array}{cccc}
\lambda: & (\mathbb{Z}_p\cup\{0\})^n\setminus\{\zero\} &\longrightarrow & \mathbb{Z}_p\times[m]\\
& X &\longmapsto & (\lambda_1(X),\lambda_2(X))
\end{array}$$ satisfying the conditions of Lemma~\ref{Z_pfanlemma} 
with parameters $n= |V({\mathcal H})|$, $m={\rm alt}_p({\mathcal H})+L$,
and $\alpha={\rm alt}_p({\mathcal H})$.
Assume that $2^{[n]}$ is equipped with a total ordering $\preceq$.
For each $X\in(\mathbb{Z}_p\cup\{0\})^n\setminus\{\zero\}$, define $\lambda(X)$ as follows.
\begin{itemize}
\item If ${\rm alt}(X)\leq {\rm alt}_p({\mathcal H},\pi)$, then let $\lambda_1(X)$ be the first nonzero coordinate of $X$ and $\lambda_2(X)={\rm alt}(X)$.
\item If ${\rm alt}(X)\geq {\rm alt}_p({\mathcal H},\pi)+1$, then in view of the definition of 
${\rm alt}_p({\mathcal H},\pi)$, there is some $\varepsilon\in\mathbb{Z}_p$
such that $E(\pi(X^\varepsilon))\neq \varnothing$.
Define 
$$c(X)=\max\left\{c(e):\; \exists\varepsilon\in\mathbb{Z}_p\mbox { such that } 
e\subseteq \pi(X^\varepsilon)\right\}$$
and $\lambda_2(X)={\rm alt}_p({\mathcal H},\pi)+c(X)$. 
Choose $X^\varepsilon$ so that 
there is at least one edge $e\in\pi (X^\varepsilon)$ with $c(e)=c(X)$ and such that
 $X^\varepsilon$ is the maximum one  having this property. By maximum, we mean
 maximum according to the total ordering $\preceq$. 
It is clear that $\varepsilon$ is defined uniquely. Now, let $\lambda_1(X)=\varepsilon$.
\end{itemize}
One can check that $\lambda$ satisfies the conditions of Lemma~\ref{Z_pfanlemma}.
Consider the chain $Z_1\subset Z_2\subset\cdots\subset Z_{n-{\rm alt}_p({\mathcal H},\pi)}$, whose existence is ensured by Lemma~\ref{Z_pfanlemma}.
Note that for each $i\in[n-{\rm alt}_p({\mathcal H},\pi)]$, we have $\lambda_2(Z_i)>{\rm alt}_p({\mathcal H},\pi)$. Consequently, $\lambda_2(Z_i)={\rm alt}_p({\mathcal H},\pi)+c(Z_i)$. Let $\lambda(Z_i)=(\varepsilon_i,j_i)$. 
Note that for each $i$, there is at least one edge 
$e_{i,\varepsilon_i}\subseteq \pi(Z_i^{\varepsilon_i})\subseteq \pi(Z_{n-{\rm alt}_p({\mathcal H},\pi)}^{\varepsilon_i})$ such that
$c(e_{i,\varepsilon_i})=j_i-{\rm alt}_p({\mathcal H},\pi)$.
For each $\varepsilon\in\mathbb{Z}_p$, define 
$U_\varepsilon=\{e_{i,\varepsilon_i}:\; \varepsilon_i=\varepsilon\}.$ 
We have the following three properties for $U_\varepsilon$'s.
\begin{itemize}
\item Since the chain $Z_1\subset Z_2\subset\cdots\subset 
	 Z_{n-{\rm alt}_p({\mathcal H},\pi)}$ is satisfying Condition~\ref{condition3} of 
	 Lemma~\ref{Z_pfanlemma}, we have 
	 $\left\lfloor{n-{\rm alt}_p({\mathcal H},\pi)\over p}\right\rfloor\leq 
	 |U_\varepsilon|\leq \left\lceil{n-{\rm alt}_p({\mathcal H},\pi)\over p}\right\rceil.$
\item The edges in $U_\varepsilon$ get distinct colors. 
	 If there are two edges $e_{i,\varepsilon}$ and $e_{i',\varepsilon}$ in $U_\varepsilon$ such that
	 $c(e_{i,\varepsilon})=c(e_{i',\varepsilon})$, then $\lambda(Z_i)=\lambda(Z_{i'}),$ 
	 which is not possible.
\item If $\varepsilon\neq \varepsilon'$, then for each $e\in U_\varepsilon$ and $f\in U_{\varepsilon'}$, 
	 we have $e\cap f=\varnothing$. It is clear because 
	 $e\subseteq\pi(Z_{n-{\rm alt}_p({\mathcal H},\pi)}^\varepsilon)$,
	 $f\subseteq\pi(Z_{n-{\rm alt}_p({\mathcal H},\pi)}^{\varepsilon'})$,
	 and 
	 $$\pi(Z_{n-{\rm alt}_p({\mathcal H},\pi)}^\varepsilon)\cap 
	 \pi(Z_{n-{\rm alt}_p({\mathcal H},\pi)}^{\varepsilon'})=\varnothing.$$
\end{itemize}
Now, it is clear that the subhypergraph ${\rm KG}^p({\mathcal H})[U_{\omega^1},\ldots,U_{\omega^p}]$ is the desired subhypergraph.
\end{proof}

%%%%%%%%%%%%%%%

Next proposition is an extension of Theorem~\ref{altercrossindex}. However, we lose some properties by this extension.

\begin{proposition}\label{Xindposet}
Let $P$ be a free ${\mathbb Z}_p$-poset and 
$$\begin{array}{rll}
\psi: P & \longrightarrow & Q_{s,p}\\
       x  &\longmapsto & (\psi_1(x),\psi_2(x))
\end{array}$$ 
be an order-preserving ${\mathbb Z}_p$-map. Then $P$ contains a chain $x_1\prec_P\cdots\prec_Px_{ k}$ such that 
\begin{itemize}
\item $k= {\rm ind}_{\Z_p}(\Delta P)+1$,
\item for each $i\in[k-1]$, $\psi_2(x_i)< \psi_2(x_{i+1})$, and
\item for each $\varepsilon\in\mathbb{Z}_p$, 
$$\left\lfloor{k\over p}\right\rfloor\leq \left|\left\{j:\; \psi_1(x_j)=\varepsilon\right\}\right|\leq \left\lceil{k\over p}\right\rceil.$$
\end{itemize}
\end{proposition}
\begin{proof}
Clearly, the map $\psi$ can be considered as a simplicial ${\mathbb Z}_p$-map from $\Delta P$ to $ \mathbb{Z}_p^{*n}\subseteq (\sigma_{p-2}^{p-1})^{*n}$. Consider the 
$(k-1)$-dimensional simplex $x_1\prec_P\cdots\prec_Px_{ k}$ in $\Delta P$, 
whose existence is ensured by Lemma~\ref{genfanlemma}. 
Set $\tau=\{\psi(x_1),\ldots,\psi(x_k)\}$. First note that we already know  
$\left\lfloor{k\over p}\right\rfloor\leq  |\tau^\varepsilon|\leq \left\lceil{k\over p}\right\rceil$
for each $\varepsilon\in\mathbb{Z}_p$. The fact that $\tau$ is a $(k-1)$-dimensional simplex in $\mathbb{Z}_p^{*n}$ implies that 
$\psi(x_i)\neq \psi(x_j)$ for each $i\neq j\in[k]$.  On the other hand, 
since $\tau$ is a simplex in $\mathbb{Z}_p^{*n}$ and $\psi$ is an order-preserving ${\mathbb Z}_p$-map, we have $\psi_2(x_i)< \psi_2(x_{i+1})$ for each $i\in[k-1]$. 
Therefore, we have 
$|\tau^\varepsilon|=|\{j:\; \psi_1(x_j)=\varepsilon\}|$
for each $\varepsilon\in\mathbb{Z}_p$, completing the proof. 
\end{proof}
Note that, for $p=2$,
since ${\rm Xind}(P)\geq {\rm ind}(\Delta P)$, 
Theorem~\ref{altercrossindex} is better than Proposition~\ref{Xindposet}.
However, we were~not able to prove that Proposition~\ref{Xindposet} 
is valid if we replace ${\rm ind}(\Delta P)$ by ${\rm Xind}(P)$.


In an unpublished paper, Meunier~\cite{unpublishedMeunier} introduced a 
generalization of Ky~Fan's lemma. He presented a version of the 
$\mathbb{Z}_q$-Fan lemma, being valid for each odd integer 
$q\geq 3$. To be more specific, he proved that if $q$ is an odd positive integer and 
$\lambda:V(T)\longrightarrow \Z_q\times[m]$ is a $\Z_q$-equivariant  labeling 
of a $\Z_q$-equivariant 
triangulation of a $(d-1)$-connected free $\Z_q$-space $T$
such that there is no edge in $T$, whose vertices are labeled with $(\varepsilon,j)$ and $(\varepsilon',j)$ with $\varepsilon\neq \varepsilon'$ and $j\in[m]$, then 
there is at least one $n$-dimensional simplex in 
$T$, whose vertices are labelled with labels 
$(\varepsilon_0,j_0),(\varepsilon_1,j_1),\ldots,(\varepsilon_n,j_n)$, 
where $\varepsilon_i\neq \varepsilon_{i+1}$ and $j_i<j_{i+1}$ for 
all $i\in\{0,1,\ldots,n-1\}$. Also, he asked the question if the result is true for
even values of $q$. This question received a positive answer owing 
to the work of B.~Hanke et~al.~\cite{Hanke2009404}.  
In both mentioned works, the proofs of the $\mathbb{Z}_q$-Fan lemma 
are built in involved construction. Here, we take the opportunity of this paper to 
propose the following generalization of this result with a short simple 
proof uisng some similar techniques as we already used in the paper.
\begin{lemma}{\rm($\mathbb{G}$-Fan lemma).}\label{Gtucker}
Let $\mathbb{G}$ be a nontrivial finite group and 
let $T$ be a free $\mathbb{G}$-simplicial complex such that ${\rm ind}_{\mathbb{G}}(T)= n$.
Assume that $\lambda:V(T)\longrightarrow \mathbb{G}\times[m]$ is a $\mathbb{G}$-equivariant  labeling such that there is no edge in $T$, whose vertices are labelled  with $(g,j)$ and $(g',j)$ with $g\neq g'\in \mathbb{G}$ and $j\in[m]$. Then there is at least one 
$n$-dimensional simplex in $T$, whose vertices are labelled with labels $(g_0,j_0),(g_1,j_1),\ldots,(g_n,j_n)$, where $g_i\neq g_{i+1}$ and $j_i<j_{i+1}$ for each $i\in\{0,1,\ldots,n-1\}$. In particular, $m\geq n+1$.
\end{lemma}
\begin{proof}
Clearly, the map $\lambda$ can be considered as a $\mathbb{G}$-simplicial map from $T$ to $\mathbb{G}^{*m}$. Note that, naturally  
each nonempty simplex $\sigma\in \mathbb{G}^{*m}$ can be identified with a vector $X=(x_1,x_2,\ldots,x_m)\in (\mathbb{G}\cup\{0\})^n\setminus\{\zero\}$. To prove the assertion, it is enough to show that there is a simplex $\sigma\in T$ such that ${\rm alt}(\lambda(\sigma))\geq n+1$. 
For a contradiction, suppose that, for each simplex $\sigma\in T$, we have ${\rm alt}(\lambda(\sigma))\leq n$. 
Define 
$$\begin{array}{lrll}
\Gamma:&V(\sd T) &\longrightarrow & \mathbb{G}\times[n]\\
		&\sigma&\longmapsto & \left(g,{\rm alt}(\lambda(\sigma)\right)),
\end{array}$$
where $g$ is the first nonzero coordinate of the vector $\lambda(\sigma)\in (\mathbb{G}\cup\{0\})^n\setminus\{\zero\}.$
One can check that $\Gamma$ is a  simplicial $\mathbb{G}$-map from $\sd T$ to $\mathbb{G}^{*n}$.
Note $\mathbb{G}^{*n}$ is an $E_{n-1} \mathbb{G}$ space. 
Consequently, ${\rm ind}_{\mathbb{G}}(\mathbb{G}^{*n})= n-1$.
This implies that ${\rm ind}_{\mathbb{G}}(T)\leq n-1$, which is a contradiction.
\end{proof}


%%%%%%%%%%%%%%%%%%%%
\subsection{\bf Hierarchy of Indices}
The aim of this subsection is to introduce some tools for the proof of Theorem~\ref{inequalities}.

Let $n,\alpha$, and $p$ be integers where $n\geq 1$, $n\geq\alpha\geq 0$, and $p$ is prime.
Define 
$$\displaystyle\Sigma_p(n,\alpha)=\Delta\left\{X\in(\mathbb{Z}_p\cup\{0\})^n:\; {\rm alt}(X)\geq \alpha+1\right\}.$$
Note that $\displaystyle\Sigma_p(n,\alpha)$ is a free simplicial $\mathbb{Z}_p$-complex
with the vertex set $$\left\{X\in(\mathbb{Z}_p\cup\{0\})^n:\; {\rm alt}(X)\geq \alpha+1\right\}.$$

\begin{lemma}\label{indsigma}
Let $n,\alpha$, and $p$ be integers where $n\geq 1$, $n\geq\alpha\geq 0$, and $p$ is prime. Then 
$${\rm ind}_{\mathbb{Z}_p}(\displaystyle\Sigma_p(n,\alpha))\geq n-\alpha-1.$$
\end{lemma}
\begin{proof}
Define
$$
\begin{array}{crcl}
\lambda:  & \sd \mathbb{Z}_p^{*n} & \longrightarrow & 
			(\mathbb{Z}_p^{*\alpha})*\left(\displaystyle\Sigma_p(n,\alpha)\right)\\
		& X					  & \longmapsto	     & 
		\left
			\{\begin{array}{cl}
			(\varepsilon,{\rm alt}(X)) &  \mbox{ if ${\rm alt}(X)\leq \alpha$}\\
			                            X    & \mbox{ if ${\rm alt}(X)\geq \alpha+1$},
			\end{array}
		\right.
\end{array}$$
where $\varepsilon$ is the first nonzero coordinate of $X$.
Clearly, the map $\lambda$ is a simplicial $\mathbb{Z}_p$-map.  
Therefore, in view of  Properties~\ref{P1} and \ref{P3} in Properties of the $\mathbb{G}$-index, we have 
$$
\begin{array}{lll}
n-1={\rm ind}_{\mathbb{Z}_p}( \sd \mathbb{Z}_p^{*n}) & \leq & {\rm ind}_{\mathbb{Z}_p}\left(\mathbb{Z}_p^{*\alpha}*\displaystyle\Sigma_p(n,\alpha)\right)\\
& \leq & {\rm ind}_{\mathbb{Z}_p}(\mathbb{Z}_p^{*\alpha})+{\rm ind}_{\mathbb{Z}_p}(\displaystyle\Sigma_p(n,\alpha))+1\\
&\leq &\alpha+{\rm ind}_{\mathbb{Z}_p}(\displaystyle\Sigma_p(n,\alpha)),
\end{array}
$$ which completes the proof.
\end{proof}
\begin{proposition}\label{inequalityI}
Let ${\mathcal F}$ be a hypergraph. For any integer $r\geq 2$ and any prime number $p\geq r$, we have
$${\rm ind}_{\mathbb{Z}_p}({\rm B}_0({\rm KG}^r({\mathcal F}),\mathbb{Z}_p))+1\geq |V({\mathcal F})|-{\rm alt}_p({\mathcal F}).$$
\end{proposition}
\begin{proof}
For convenience, let $|V({\mathcal F})|=n$ and $\alpha={\rm alt}_p({\mathcal F})$.
Let $\pi:[n]\longrightarrow V({\mathcal F})$ be a bijection such that 
${\rm alt}_p({\mathcal F},\pi)={\rm alt}_p({\mathcal F})$. 
Define 
$$
\begin{array}{lrll}
\lambda:& \Sigma_p(n,\alpha)& \longrightarrow & \sd{\rm B}_0({\rm KG}^r({\mathcal F}),\mathbb{Z}_p))\\
 & X&\longmapsto & \{\omega^1\}\times U_1\cup\cdots\cup \{\omega^p\}\times U_p,
\end{array}
$$  where $U_i=\{e\in E({\mathcal F}):\; e\subseteq \pi(X^{\omega^i})\}.$
One can see that $\lambda$ is a  simplicial $\mathbb{Z}_p$-map.
Consequently, 
\[{\rm ind}_{\mathbb{Z}_p}({\rm B}_0({\rm KG}^r({\mathcal F}),\mathbb{Z}_p))\geq 
{\rm ind}_{\mathbb{Z}_p}(\Sigma_p(n,\alpha))\geq n-{\rm alt}_p({\mathcal F})-1.\qedhere\]
\end{proof}

\begin{proposition}\label{inequalityII}
Let ${\mathcal H}$ be an $r$-uniform hypergraph and $p\geq r\geq 2$ be a prime number.
Then
$$ {\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p\geq  {\rm ind}_{\mathbb{Z}_p}(\Delta{\rm Hom}(K^r_p,{\mathcal H}))+p\geq  {\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1.$$
\end{proposition}
\begin{proof}
Since we already know that 
${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))\geq  {\rm ind}_{\mathbb{Z}_p}(\Delta{\rm Hom}(K^r_p,{\mathcal H}))$, 
to prove the assertion, it suffices to show that 
${\rm ind}_{\mathbb{Z}_p}(\Delta{\rm Hom}(K^r_p,{\mathcal H}))+p\geq  {\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))+1.$
To this end, let 
$$\begin{array}{llll}
\lambda: & \sd {\rm B_0}({\mathcal H},\mathbb{Z}_p) & \longrightarrow & \left(\sd\sigma_{p-2}^{p-1}\right)*\displaystyle\left(\Delta{\rm Hom}(K^r_p,{\mathcal H})\right)\\
\end{array}$$
be a map such that for each vertex 
$\tau=\displaystyle\bigcup_{i=1}^p\left(\{\omega^i\}\times U_i\right)$ of $\sd {\rm B_0}({\mathcal H},\mathbb{Z}_p)$, we define  $\lambda(\tau)$  as follows.
\begin{itemize}
\item If $U_i\neq\varnothing$ for each $i\in[p]$, then $\lambda(\tau)=\tau.$
\item If $U_i=\varnothing$ for some $i\in[p]$, then 
$$\lambda(\tau)=\{\omega^i\in\mathbb{Z}_p:\; U_i=\varnothing\}.$$
\end{itemize} 
One can check that the map $\lambda$ is a simplicial $\mathbb{Z}_p$-map. Also,
since $\sigma_{p-2}^{p-1}$ is a free simplicial $\mathbb{Z}_p$-complex of 
dimension $p-2$, we have ${\rm ind}_{\mathbb{Z}_p}(\sigma_{p-2}^{p-1})\leq p-2$
(see Property~\ref{P5} in Properties of the $\mathbb{G}$-index).
This implies that 
$$
\begin{array}{lll}
 {\rm ind}_{\mathbb{Z}_p}({\rm B_0}({\mathcal H},\mathbb{Z}_p))& \leq &  {\rm ind}_{\mathbb{Z}_p}\left(\left(\sd\sigma_{p-2}^{p-1}\right)*
\left(\Delta{\rm Hom}(K^r_p,{\mathcal H})\right)\right)\\
& \leq &{\rm ind}_{\mathbb{Z}_p}(\sigma_{p-2}^{p-1})+ {\rm ind}_{\mathbb{Z}_p}(\Delta{\rm Hom}(K^r_p,{\mathcal H}))+1\\
&\leq & p-1+{\rm ind}_{\mathbb{Z}_p}(\Delta{\rm Hom}(K^r_p,{\mathcal H})),
\end{array}
$$ which completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\bf Proofs of Theorem~\ref{maincolorfulindex} and Theorem~\ref{inequalities}}\label{sec:proofs}
Now, we are ready to prove Theorem~\ref{maincolorfulindex} and Theorem~\ref{inequalities}.\\


\begin{proof}[Proof of Theorem~{\rm \ref{maincolorfulindex}}]{\bf Part (i).}
For convenience, let ${\rm ind}_{\mathbb{Z}_p}({\rm B}_0({\mathcal H},{\mathbb{Z}_p}))=t$.
One can readily check that the map 
$$
\begin{array}{crcl}
\Gamma:  &\mathbb{Z}_p\times V({\mathcal H}) & \longrightarrow & \mathbb{Z}_p\times [L]\\
		  & (\varepsilon,v)		  			   & \longmapsto	     & (\varepsilon,c(v))
\end{array}$$ is a simplicial $\mathbb{Z}_p$-map from ${\rm B}_0({\mathcal H},{\mathbb{Z}_p})$ to $(\sigma^{p-1}_{r-2})^{*L}$. Therefore, in view of Lemma~\ref{genfanlemma}, there is a $t$-dimensional simplex 
$\sigma=\ds\bigcup_{i=1}^p(\{\omega^i\} \times U_i)\in {\rm B}_0({\mathcal H},{\mathbb{Z}_p})$ such that $\tau=\Gamma(\sigma)$ is also $t$-dimensional and moreover,  
$\lfloor{t+1\over p}\rfloor\leq |\tau^\varepsilon|\leq\lceil{t+1\over p}\rceil$ 
for each $\varepsilon\in \mathbb{Z}_p$.
Since $\sigma$ is a $t$-dimensional simplex in ${\rm B}_0({\mathcal H},{\mathbb{Z}_p})$, 
\begin{itemize} 
\item $\ds\sum_{i=1}^p |U_i|=t+1$ and 
\item ${\mathcal H}[U_1,\ldots,U_p]$ is a complete $r$-uniform $p$-partite subhypergraph
of $\mathcal{H}$. 
\end{itemize}
In view of the definition of $\Gamma$ and 
since $\tau=\Gamma(\sigma)$ is also a $t$-dimensional simplex, we must have 
$|U_i|=|c(U_i)|=|\tau^{\omega^i}|$ for each $i\in[p]$. 
Now, it is clear that ${\mathcal H}[U_1,\ldots,U_p]$ is the desired subhypergraph, completing the proof in this part.\\
%Moreover,
%since every color {\color{red} appears} in at most $r-1$ number of $U_i$'s,  we have
%$$L\geq {{\rm ind}_{\mathbb{Z}_p}({\rm B}_0({\mathcal H},{\mathbb{Z}_p}))+1\over r-1}.$$

\noindent{\bf Part (ii).}
First note that since $p\leq \omega ({\mathcal H})$, the $\Z_p$-poset ${\rm Hom}(K^r_p,{\mathcal H}))$ is not empty.  
For convenience, let ${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))=t$. 
Let $F$ be {\it the face poset of $(\sigma_{r-2}^{p-1})^{*L}$}, i.e., 
the poset consisting of all nonempty simplices of 
$(\sigma_{r-2}^{p-1})^{*L}$  ordered by inclusion. Since $(\sigma_{r-2}^{p-1})^{*L}$ is a free
simplicial $\Z_p$-complex, 
one can readily check that $F$ is a free $\Z_p$-poset. Also, it is clear that
the ground set of $F$ is the same as the vertex set of 
$\sd\left((\sigma_{r-2}^{p-1})^{*L}\right)$.
%By abuse of notation, let $\sd\left((\sigma_{r-2}^{p-1})^{*L}\right)$ also denotes  the {\it face poset of $(\sigma_{r-2}^{p-1})^{*L}$}, i.e., the $\Z_p$-poset consisting of all nonempty simplices of $(\sigma_{r-2}^{p-1})^{*L}$  ordered by inclusion.  
Now, define the map $$\lambda:{\rm Hom}(K^r_p,{\mathcal H})\longrightarrow F$$
such that for each $(U_1,\ldots,U_p)\in {\rm Hom}(K^r_p,{\mathcal H})$, 
$$\lambda(U_1,\ldots,U_p)=\{\omega^1\}\times c(U_1) \cup\cdots\cup \{\omega^p\}\times c(U_p).$$


\noindent{\bf Claim.} There is a $p$-tuple $(U_1,\ldots,U_p)\in {\rm Hom}(K^r_p,{\mathcal H})$
such that for $\tau=\lambda(U_1,\ldots,U_p)$, we have $l(\tau)\geq t+p$.

\begin{proof}[Proof of Claim] For sake a contradiction, suppose that for each $\tau\in {\rm Im}(\lambda)$, we have $l(\tau)\leq t+p-1$.
%Note that $\sd\left((\sigma_{r-2}^{p-1})^{*L}\right)$ is considered as a free $\mathbb{Z}_p$-poset ordered by inclusion.
One can readily check that $\lambda$ is an  order-preserving $\mathbb{Z}_p$-map.
Clearly, for each $\tau\in {\rm Im}(\lambda)$, we have $h(\tau)=\ds\min_{\varepsilon\in\mathbb{Z}_p}|\tau^\varepsilon|\geq 1$ and consequently, $l(\tau)\geq p$. Now, define  
$$\begin{array}{ccrcl}
\Gamma &: & {\rm Im}(\lambda) & \longrightarrow & Q_{t-1,p}\\ \\
		& & \tau & \longmapsto & \left(s(\bar\tau), l(\tau)-p+1\right), 
\end{array}
$$ where $\bar{\tau}=\bigcup\limits_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h(\tau)\}} \tau^\varepsilon\in W$. 
%$$\bar{\tau}=	        \bigcup\limits_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h(\tau)\}} \tau^\varepsilon\in W\quad {\rm and }\quad \Gamma(\tau)=\left(s(\bar\tau), l(\tau)-p+1\right).$$
Clearly, since $s(-)$ is  a $\mathbb{Z}_p$-equivariant map,  $\Gamma$ is a $\mathbb{Z}_p$-equivariant map as well. 	        
One can see that the map $\Gamma:{\rm Im}(\lambda)\longrightarrow Q_{t-1,p}$ is an order-preserving $\mathbb{Z}_p$-map. 
To this end, in view of the definition of the ordering in $Q_{t-1,p}$, it suffices to show that 
if $\Gamma(\tau)=(\varepsilon_1,\beta)$ and $\Gamma(\tau')=(\varepsilon_2,\beta)$ for some  $\tau\subsetneq\tau' \in {\rm Im}(\lambda)$, then $\varepsilon_1=\varepsilon_2$. 
For a contradiction, suppose that there are $\tau,\tau' \in {\rm Im}(\lambda)$
such that $\tau\subsetneq\tau'$, $\Gamma(\tau)=(\varepsilon_1,\beta)$, and 
$\Gamma(\tau')=(\varepsilon_2,\beta)$, where $\varepsilon_1\neq \varepsilon_2$. 
Clearly, in view of definition of $\Gamma$, we have $l(\tau)=l(\tau')=\beta+p-1$.
On the other hand, we know that
	 $$ l(\tau)=p\cdot h(\tau)+|\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|>h(\tau)\}|
	 \mbox{ and } l(\tau')=p\cdot h(\tau')+|\{\varepsilon\in\mathbb{Z}_p:\; 
	 |{\tau'}^\varepsilon|>h(\tau')\}|.$$  
The facts that  $l(\tau)=l(\tau')$ and  
$$\max\left\{|\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|>h(\tau)\}|, |\{\varepsilon\in\mathbb{Z}_p:\; 
	 |{\tau'}^\varepsilon|>h(\tau')\}|\right\}\leq p-1$$
imply that  $h(\tau)=h(\tau')$ and 
\begin{equation}\label{equationnew}
|\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|>h\}|
	 = |\{\varepsilon\in\mathbb{Z}_p:\; 
	 |{\tau'}^\varepsilon|>h\}|,
\end{equation} 
where we set $h=h(\tau)=h(\tau')$. 
In view of 
	 $$\varepsilon_1=s(\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h\}} \tau^\varepsilon)\neq s(\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\}} {\tau'}^\varepsilon)=\varepsilon_2,$$
we must have $$\displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |\tau^\varepsilon|=h\}} \tau^\varepsilon\neq \displaystyle\bigcup_{\{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\}} {\tau'}^\varepsilon.$$ 
Also, $\tau\subsetneq \tau'$ and $\ds\min_{\varepsilon\in \mathbb{Z}_p}|\tau^\varepsilon|=\ds\min_{\varepsilon\in \mathbb{Z}_p}|{\tau'}^\varepsilon|$ implies that  
	 $$
	 \left\{\varepsilon\in\mathbb{Z}_p:\; |{\tau'}^\varepsilon|=h\right\} \subsetneq \left\{\varepsilon\in\mathbb{Z}_p:\; |{\tau}^\varepsilon|=h\right\},$$ which contradicts 
Equality~\ref{equationnew}.

Therefore, since both $\Gamma$ and $\lambda$ are order-preserving $\Z_p$-maps,  
$$\Gamma\circ\lambda:{\rm Hom}(K^r_p,{\mathcal H})\longrightarrow Q_{t-1,p}$$ is an 
order-preserving $\mathbb{Z}_p$-map as well, contradicting the fact that ${\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))=t$.
\end{proof}
Amongst all $p$-tuples, whose existence are ensured by Claim, 
choose a minimal one, say 
$T=(V_1,\ldots,V_p)\in {\rm Hom}(K^r_p,{\mathcal H})$. 
First note that since $(V_1,\ldots,V_p)$ is in ${\rm Hom}(K^r_p,{\mathcal H})$, the 
subhypergraph ${\mathcal H}[V_1,\ldots,V_p]$ is a complete 
$r$-uniform $p$-partite hypergraph. 
Set $\tau=\lambda(T)$. In view of the minimality of $T$, we clearly have 
$$\displaystyle\sum_{i=1}^{p}|V_i|=|T|=|\tau|=l(\tau)=t+p.$$ 
In view of the definition of $\lambda(-)$, it implies that $|V_i|=|c(V_i)|=|\tau^{\omega^i}|$ for each $i\in[p]$. 
Also, the equalities $|\tau|=l(\tau)=t+p$ imply that $\left\lfloor{t+p\over p}\right\rfloor\leq |\tau^{\omega^i}|\leq \left\lceil{t+p\over p}\right\rceil$. Therefore,  ${\mathcal H}[V_1,\ldots,V_p]$ is the desired complete 
$r$-uniform $p$-partite subhypergraph, completing the proof.  
%Similar to the proof of Part (i), since every color {\color{red} appears} in at most $r-1$ number of $V_i$'s,  we have $$L\geq {{\rm Xind}_{\mathbb{Z}_p}({\rm Hom}(K^r_p,{\mathcal H}))+p\over r-1}.$$
\end{proof}

\begin{proof}[Proof of Theorem~{\rm\ref{inequalities}}]
It has already be noted that
$|V({\mathcal F})|-{\rm alt}_p({\mathcal F})
\geq  {\rm cd}_p({\mathcal F})$ for any hypergraph ${\mathcal F}$.
Therefore, the proof follows by  Proposition~\ref{inequalityI} and Proposition~\ref{inequalityII}.
\end{proof}

\section*{Acknowledgments}
This research was in part supported by a grant from IPM (No. 95050014). 
I would like to acknowledge Professor Fr\'ed\'eric~Meunier for interesting 
discussions about the paper and his invaluable comments. 
Also, I would like to thank Professor Hossein~Hajiabolhassan and Mrs~Roya~Abyazi~Sani 
for their useful comments. Moreover, the author is grateful to the anonymous  referee 
for carefully reading the paper and giving several constructive comments that helped 
to improve the presentation of the paper.

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%\bibliographystyle{plain}
%\bibliography{MyReferences}

\def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{99}

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M.~{Alishahi} and H.~{Hajiabolhassan}.
\newblock {A Generalization of Gale's lemma}.
\newblock {\em ArXiv e-prints}, \arxiv{1607.08780v1}, July 2016.


\bibitem{AliHajiMeu2016}
M.~{Alishahi}, H.~{Hajiabolhassan}, and F.~{Meunier}.
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