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\title{The Ramsey number of loose triangles and quadrangles in
hypergraphs}

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\bigskip


\author{
 Andras Gy\'{a}rf\'{a}s\thanks{Supported in part by OTKA grant K68322.}\\
\small Computer and Automation Research Institute\\[-0.8ex]
\small Hungarian Academy of Sciences\\[-0.8ex]
\small Budapest, P.O. Box 63, Hungary, H-1518\\
%\small and\\
%\small School of Mathematics\\ [-0.8ex]
%\small Institute for Research in Fundamental Sciences\\[-0.8ex]
%\small Tehran, 19395-5746, Iran\\
\small \texttt{gyarfas@sztaki.hu}\\
\and
 Ghaffar Raeisi\\
\small Department of Mathematical Sciences\\[-0.8ex]
\small Isfahan University of Technology\\[-0.8ex]
\small Isfahan, 84156-83111, Iran\\
\small \texttt{g.raeisi@math.iut.ac.ir}
}
% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

%\bigskip
%\vspace{2cm}

\date{\dateline{Apr 5, 2011}{May 11, 2012}{Jun 6, 2012}\\
\small Mathematics Subject Classifications: 05C15, 05C55, 05C65}

\begin{document}

\maketitle

\begin{abstract}
Asymptotic values of hypergraph Ramsey numbers for loose
cycles (and paths) were determined recently. Here we determine
some of them exactly, for example the 2-color hypergraph Ramsey
number of a $k$-uniform loose 3-cycle or 4-cycle:
$R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$ and
$R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$ (for $k\ge 3$). For more
than $3$ colors we could prove only that
$R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8$.
Nevertheless, the $r$-color Ramsey number of triangles for
hypergraphs are much smaller than for graphs: for $r\ge 3$,
$$r+5\le R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots,
\mathcal{C}_3^3)\le 3r$$
  % keywords are optional
\bigskip\noindent \textbf{Keywords:} Hypergraph Ramsey Number, Loose Cycle, Loose Path.
\end{abstract}

\section{ \large Introduction}

\hspace{.5 cm}In this paper, we consider the problem of finding
the 2-color Ramsey number for uniform hypergraph paths and cycles.
There are several natural definitions for a cycle and a path in a
uniform hypergraph. The one we focus on here is called loose. The
$k$-uniform {\it loose cycle} $\mathcal{C}_n^k$ (shortly, a {\it
cycle of length $n$}),  is the hypergraph with vertex set
$\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ hyperedges
$e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$, $i=0,1,\ldots, n-1$, where
we use mod $n$ arithmetic and adding a number $t$ to a set
$H=\{v_1,v_2,\ldots, v_k\}$ means a shift, i.e. the set obtained
by adding $t$ to subscripts of each element of $H$. Similarly, the
$k$-uniform {\it loose path} $\mathcal{P}_n^k$ (simply, a {\it
path of length $n$}), is the hypergraph with vertex set
$\{v_1,v_2,\ldots,v_{n(k-1)+1}\}$  and with the set of $n$
hyperedges $e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$, $i=0,1,\ldots,
n-1$. Also by a {\it star} $\mathcal{S}_n^k$ we mean the
$k$-uniform hypergraph with vertex set
$\{v,v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges
$e_i=\{v\}\cup\{v_1,v_2,\ldots, v_{k-1}\}+i(k-1)$, $i=0,1,\ldots,
n-1$. For $k=2$ we get the definition of the cycle $C_n$, the path
$P_n$ and the star $K_{1,n}$ with $n$ edges. A classical result in
graph Ramsey theory (see \cite{Ramsey number of paths}) states
that $R(P_n,P_m)=n+\big\lfloor\frac{m+1}{2}\big\rfloor,$ where
$n\geq m\geq 1$. Also it is well-known (see
\cite{Ramsy number of
cycles,Ramsey number of cycle-path})
that for every $n\geq m$ and
even $m$, $R(P_n,C_m)=R(P_n,P_m)=n+\frac{m}{2}$.

\bigskip
For given $k$-uniform hypergraphs ${\mathcal H}_1,
\mathcal{H}_2,\ldots,\mathcal{H}_t$ the {\it Ramsey number}
$R(\mathcal{H}_1, \mathcal{H}_2,\ldots,\mathcal{H}_t)$ is the
smallest integer $N$ such that in every $t$-coloring of the edges
of the complete $k$-uniform hypergraph $\mathcal{K}_N^k$ there is
a monochromatic copy of $\mathcal{H}_i$ in color $i$, for some
$i$, $1\leq i\leq t$. It was proved \cite{HLR} that
$R(\mathcal{C}^3_n, \mathcal{C}^3_n)$ is asymptotically equal to
$\frac{5n}{2}$. Subsequently, Gy\'{a}rf\'{a}s et. al. in
\cite{GSS} extended this result to the $k$-uniform loose cycles
and proved that $R(\mathcal{C}^k_n, \mathcal{C}^k_n)$ is
asymptotically equal  to $\frac{1}{2}(2k-1)n$.
 Here we determine the exact value of the 2-color Ramsey
number of a $k$-uniform loose triangle and quadrangle,
$R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$ (Theorem \ref{Ramsey
number of triangles}) and
$R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$ (Theorem \ref{Ramsey
number of C4}). We observe that the $r$-color Ramsey number of
loose triangles in $3$-uniform hypergraphs is linear, in contrast
to the graph case, when it is at least exponential (Proposition
\ref{rcolor}).

\bigskip
 Perhaps the following lower
bound always gives the exact value of the 2-color hypergraph
Ramsey numbers for loose paths and cycles. This is certainly the
case for all results of this paper, therefore we shall always
prove just the claimed upper bounds for the Ramsey numbers. During
this paper, for a 2-edge coloring of a uniform hypergraph
$\mathcal{H}$, say red and blue,  we denote by $\mathcal{F}_{red}$
and $\mathcal{F}_{blue}$ the induced hypergraph on edges of color
red and blue, respectively.
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\medskip
\begin{lemma}\label{lower bound}
For every $n\geq m\geq 2$ and $k\geq 3$,
$(k-1)n+\lfloor\frac{m+1}{2}\rfloor$ is a lower bound for both
$R(\mathcal{P}^k_n,\mathcal{P}^k_m)$ and
$R(\mathcal{P}^k_n,\mathcal{C}^k_m)$. Moreover,
$R(\mathcal{C}^k_n,\mathcal{C}^k_m)\geq
(k-1)n+\lfloor\frac{m+1}{2}\rfloor-1$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }To prove that
$(k-1)n+\lfloor\frac{m+1}{2}\rfloor$ is a lower bound for both
Ramsey numbers $R(\mathcal{P}^k_n,\mathcal{P}^k_m)$ and
$R(\mathcal{P}^k_n,\mathcal{C}^k_m)$, we exhibit a 2-coloring of
the edges of the complete $k$-uniform hypergraph on
${(k-1)n+\lfloor\frac{m+1}{2}\rfloor}-1$ vertices such that
$\mathcal{P}_n^k\nsubseteq \mathcal{F}_{red}$ and
$\mathcal{P}_m^k\nsubseteq \mathcal{F}_{blue}$ (also
$\mathcal{C}_m^k\nsubseteq \mathcal{F}_{blue}$). For this purpose,
Partition the vertex set into parts $A$ and $B$, where
$|A|=(k-1)n$ and $|B|=\lfloor\frac{m+1}{2}\rfloor-1$. We color all
edges that contain a vertex of $B$ blue, and the rest red. Now,
this coloring can not contain a red copy of $\mathcal{P}_n^k$,
since any such copy must have all vertices in $A$ and
$|A|=(k-1)n$. Clearly a longest blue path (also a longest blue
cycle) has length at most $m-1$, as claimed.
 By the same argument and
considering  parts $A$ and $B$ with  $|A|=(k-1)n-1$ and
$|B|=\lfloor\frac{m+1}{2}\rfloor-1$, we obtain that
$(k-1)n+\lfloor\frac{m+1}{2}\rfloor-1\leq
R(\mathcal{C}^k_n,\mathcal{C}^k_m)$, which completes the proof.
$\hfill \square$

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\medskip


\section{\large Ramsey number of short paths and cycles}

\medskip
%\hspace{.5 cm}In this section, we prove that  for $k\geq 3$, $R(\mathcal{P}^k_3,\mathcal{P}^k_3)=R(\mathcal{C}^k_3,\mathcal{C}^k_3)+1=3k-1$
%and also $R(\mathcal{P}^k_4,\mathcal{P}^k_4)=R(\mathcal{C}^k_4,\mathcal{C}^k_4)+1=4k-2$.
%Moreover, the 3-color Ramsey number of $\mathcal{C}^k_3$ is probably $3k-1$ but we could prove this only for $k=3$.

\hspace{.5 cm}The following lemma shows an easy but useful
property of monochromatic cycles in uniform hypergraphs on
$R(\mathcal{P}^k_n,\mathcal{P}^k_m)$ vertices.

\medskip
\begin{lemma}\label{No Cn}
Let $n\geq m\geq 3$ and
$\mathcal{K}^k_{(k-1)n+\lfloor\frac{m+1}{2}\rfloor}$ be 2-edge
colored red and blue. If $\mathcal{C}_{n}^k\subseteq
\mathcal{F}_{red}$ then either $\mathcal{P}_n^k\subseteq
\mathcal{F}_{red}$ or $\mathcal{P}_m^k\subseteq
\mathcal{F}_{blue}$. Also, if $\mathcal{C}_{n}^k\subseteq
\mathcal{F}_{red}$ then either $\mathcal{P}_n^k\subseteq
\mathcal{F}_{red}$ or $\mathcal{C}_m^k\subseteq
\mathcal{F}_{blue}$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }Let $e_i=\{v_1,v_2,\ldots, v_k\}+i(k-1)$
(mod $n$), $i=0,1,\ldots, n-1$, be the edges of
$\mathcal{C}_{n}^k\subseteq \mathcal{F}_{red}$ and
$W=\{x_1,x_2,\ldots,x_{\lfloor\frac{m+1}{2}\rfloor}\}$
 be the set of the remaining vertices. Let $1\leq i\leq m-1$ and set $e_{0}^{\prime}=(e_{0}\setminus\{v_{1}\})\cup\{x_{1}\}$,  $e_i^{\prime}=(e_i\setminus \{v_{(i+1)(k-1)+1}\})\cup\{x_{\frac{i+2}{2}}\}$
 if $i$ is even and $e_i^{\prime}=(e_i\setminus\{v_{i(k-1)+1}\})\cup\{x_{\frac{i+1}{2}}\}$ if $i$ is odd. If one of $e_i^{\prime}$ is red,  we have a monochromatic $\mathcal{P}_n^k\subseteq \mathcal{F}_{red}$,
 otherwise $e_0^{\prime}e_1^{\prime}\ldots e^{\prime}_{m-1}$ form a  $\mathcal{P}_m^k\subseteq \mathcal{F}_{blue}$,
 which completes the proof of the first part.
For the second part, for even $m$ replace the hyperedges
$e^{\prime}_{0}$ and $e^{\prime}_{m-1}$ by
$(e_{0}\setminus\{v_{k}\})\cup\{x_{1}\}$ and
$(e_{n-1}\setminus\{v_{(n-1)(k-1)+1}\})\cup\{x_{\frac{m}{2}}\}$,
respectively,  and for odd $m$ replace $e^{\prime}_{0}$ by
$(e_{0}\setminus\{v_{k-1},v_k\})\cup\{x_1,x_{\frac{m+1}{2}}\}$. By
a similar argument, $e_0^{\prime}e_1^{\prime}\ldots
e^{\prime}_{m-1}$ form a $\mathcal{C}_m^k\subseteq
\mathcal{F}_{blue}$, which completes the proof.
$\hfill \square$
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\medskip
Another tool is a simple but useful folkloristic remark.

\bigskip
\begin{remark}\label{start}
If a $k$-uniform complete hypergraph is 2-colored and both colors
are used at least once, then there are two edges of distinct
colors intersecting in $k-1$ vertices.
\end{remark}\vspace{.1 cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }Select a red hyperedge $e$ and a blue
hyperedge $f$ with maximum intersection and, on contrary, suppose
that $m=|e\cap f|<k-1$. Let $g$ be an edge that contains $e\cap f$
and intersects $e\setminus f$ in $\lfloor {k-m\over 2}\rfloor$
vertices and intersects $f\setminus e$ in $\lceil {k-m\over
2}\rceil$ vertices. Now, $g,e$ or $g,f$ contradicts the choice of
$e$ and $f$. $\hfill \square$
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\bigskip
\begin{corollary}\label{P2 exists}
$R(\mathcal{P}^k_2,\mathcal{P}^k_2)=2k-1$.
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }Let $\mathcal{K}_{2k-1}^k$ be 2-edge
colored red and blue. We may assume that $\mathcal{F}_{red}$ and
$\mathcal{F}_{blue}$ are both nonempty. By Remark \ref{start},
assume that $e=\{v_1,v_2.\ldots ,v_k\}\in \mathcal{F}_{red}$ and
$e^{\prime}=\{v_2,v_3,\ldots, v_{k+1}\}\in \mathcal{F}_{blue}$.
Let $W$ be the set of the remaining vertices of
$\mathcal{K}_{2k-1}^k$. If the $k$-set
$e^{\prime\prime}=\{v_1,v_{k+1}\}\cup W$ is red then
$e,e^{\prime\prime}$ is a red copy of $\mathcal{P}^k_2$, otherwise
$e^{\prime},e^{\prime\prime}$ is a blue copy of $\mathcal{P}^k_2$.
$\hfill \square$
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\bigskip
\begin{theorem}\label{Ramsey number of triangles}
For every $k\geq 3$, $R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }Suppose indirectly that the edges of
$\mathcal{K}_{3k-2}^k$ can be colored red and blue without
monochromatic $\mathcal{C}^k_3$. Then  $\mathcal{F}_{red}$ and
$\mathcal{F}_{blue}$ are both nonempty, using Remark \ref{start},
select edges $e=\{v_1,v_2,\ldots v_k\}\in \mathcal{F}_{red}$ and
$e^{\prime}=\{v_2,v_3,\ldots v_{k+1}\}\in \mathcal{F}_{blue}$. Let
$v\in V(\mathcal{K}_{3k-2}^k)\setminus (e\cup e^{\prime})$ be an
arbitrary vertex and let $W$ be the set of the remaining vertices.
Note that $|W|=2k-4$. Partition the vertices of $W$ into two sets
$A$ and $B$ with $|A|=|B|=k-2$. The case $k=3$ can be easily
settled as follows. Since there is no monochromatic
$\mathcal{C}^k_3$, w.l.o.g. $e_1=\{v,v_2,w_1\}$ is red,
$e_2=\{v,v_3,w_2\}$ is blue, where $W=\{w_1,w_2\}$. Now, if
$f=\{v,v_1,v_4\}$ is red then $e,f,e_1$ is red $\mathcal{C}^k_3$,
otherwise, $e^{\prime},f, e_2$ is blue $\mathcal{C}^k_3$. So we
may assume that $k\ge 4$ and we prove the following claim.

\bigskip
\noindent \textbf{Claim. }Let $T\subseteq A$, $S\subseteq B$ such
that $|T|=t\ge 1$, $|S|=s$ and $s+t=k-2$. Let $t_1\in T$ and
$s_1\in B\setminus S$ be arbitrary vertices. If $e_1=\{v,v_2\}\cup
T\cup S$ is red, then $\{v,v_i\}\cup (T\setminus t_1)\cup (S\cup
\{s_1\})$ is red for any $v_i\in \{v_3,\dots,v_k\}$.

\bigskip \noindent{\it Proof. } Let $T^{\prime}=A\setminus T$,  $S^{\prime}=B\setminus S$.
Note that $|T^{\prime}|=s$ and $|S^{\prime}|=t$. We may assume
that the hyperedge $e_2=\{v_1,v_{k+1},v\}\cup T^{\prime}\cup
(S^{\prime}\setminus s_1)$ is blue, otherwise $e,e_1,e_2$ would
form a red $\mathcal{C}^k_3$. Now, If the hyperedge
$e_3=\{v,v_i\}\cup (T\setminus t_1)\cup (S\cup \{s_1\})$, $v_i\in
\{v_3,\dots,v_k\}$,  is red as claimed, otherwise
$e^{\prime},e_2,e_3$ would form a  blue $\mathcal{C}^k_3$.

\bigskip Now we return to the proof of the theorem. Without loss of generality, we may assume that the
edge $e_1=\{v,v_2\}\cup A$ is red. Using the claim successively
$k-2$ times switching $v_3$ and $v_4$, we obtain that $e_1$ and
$\{v,v_3\}\cup B$ are both red, giving a red $\mathcal{C}^k_3$
with $e$ and the proof is finished.
$\hfill\square$
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\vspace{.3 cm} Theorem \ref{Ramsey number of triangles} gives the
following corollary.

\medskip
\begin{corollary}\label{ramsey number of P3}
For every $k\geq 3$,
$R(\mathcal{P}^k_3,\mathcal{P}^k_3)=R(\mathcal{C}^k_3,\mathcal{P}^k_3)=3k-1$
and $R(\mathcal{S}^k_3,\mathcal{S}^k_3)=3k-2$.
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Proof. }To prove
$R(\mathcal{P}^k_3,\mathcal{P}^k_3)\leq 3k-1$, suppose that the
edges of $\mathcal{K}_{3k-1}^k$ are arbitrary colored red and
blue. By Theorem \ref{Ramsey number of triangles}, we must have a
monochromatic copy of $\mathcal{C}^k_3$. Without loss of
generality, we may assume that $\mathcal{C}^k_3\subseteq
\mathcal{F}_{red}$. Using Lemma \ref{No Cn}, we obtain that
$\mathcal{P}^k_3\subseteq \mathcal{F}_{red}$ or
$\mathcal{P}^k_3\subseteq \mathcal{F}_{blue}$, which shows that
$R(\mathcal{P}^k_3,\mathcal{P}^k_3)\leq 3k-1$. By the same
argument, one can easily obtain that
$R(\mathcal{P}^k_3,\mathcal{C}^k_3)\leq 3k-1$. To see
$R(\mathcal{S}^k_3,\mathcal{S}^k_3)\leq3k-2$, let the edges of
$\mathcal{K}_{3k-2}^k$ be arbitrary colored red and blue. By
Theorem \ref{Ramsey number of triangles}, we have a monochromatic,
say red, copy of  $\mathcal{C}^k_3$. Assume that
$e_1=\{v,v_1,\ldots, v_{k-2},u\}$, $e_2=\{u,u_1,\ldots,
u_{k-2},w\}$ and $e_3=\{w,w_1,\ldots ,w_{k-2},v\}$ are the
hyperedges of the copy of $\mathcal{C}_3^k$ and $T=\{t\}$ be the
remaining vertex of $\mathcal{K}_{3k-2}^k$. If one of the
hyperedges $e_1^{\prime}=\{t,w,v_1,\ldots, v_{k-2}\}$,
$e_2^{\prime}=\{t,u,w_1,\ldots, w_{k-2}\}$ or
$e_3^{\prime}=\{t,v,u_1,\ldots, u_{k-2}\}$ is red, then we have a
red copy of $\mathcal{S}_3^k\subseteq \mathcal{F}_{red}$,
otherwise $e_1^{\prime}e_2^{\prime}e_3^{\prime}$ form a
$\mathcal{S}_3^k\subseteq \mathcal{F}_{blue}$, which completes the
proof.
 $\hfill \square$
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\bigskip
To determine $R(\mathcal{C}^k_{4},\mathcal{C}^k_{4})$ for
$k\geq3$, we need the following lemma.
%Assume that $n\geq 2k+2$ and $\mathcal{G}=\mathcal{K}^k_n$ is 2-edge colored red and blue. We say that $\mathcal{G}$ has {\it partition
%property} if in any partition of $V(\mathcal{G})$ into two parts $V_1$ and $V_2$ with $|V_i|\geq k+1$, one of the
%parts span a monochromatic hypergraph.

\bigskip
\begin{lemma}\label{partition Property}
If $\mathcal{K}^k_{4k-3}$ is 2-edge colored red and blue without
monochromatic $\mathcal{C}_4^k$, then there exist two pairs of
red-blue hyperedges $e,e^{\prime}$ and $f,f^{\prime}$ such that
$(e\cup e^{\prime})\cap (f\cup f^{\prime})=\emptyset$.
\end{lemma}
\noindent\textbf{Proof. }Set $V=V(\mathcal{K}^k_{4k-3})$. Since
there is no monochromatic $\mathcal{C}^k_{4}$, $\mathcal{F}_{red}$
and $\mathcal{F}_{blue}$ are both nonempty. By Remark \ref{start},
select $e=\{v_1,v_2,\ldots,v_{k}\}\in \mathcal{F}_{red}$ and
$e^{\prime}=\{v_2,v_3,\ldots,v_{k+1}\}\in \mathcal{F}_{blue}$ and
let $W$ denote the set of vertices uncovered by $e\cup
e^{\prime}$. If $W$ is not monochromatic, we can select red-blue
hyperedges $f,f^{\prime}$ in $W$ which clearly $(e\cup
e^{\prime})\cap (f\cup f^{\prime})=\emptyset$. Thus let $W$ span a
monochromatic hypergraph, w.l.o.g blue. Let $W_1,W_2$ be disjoint
subsets of $W$ with cardinality $k-1$ and $w\in W\setminus
(W_1\cup W_2)$ and $h_i=W_i\cup \{w\}$ ($i=1,2$).  We may assume
that $g_1=\{v_{k+1}\}\cup W_1$ is blue. Indeed, otherwise consider
$g_2=\{v_1\}\cup W_2$.
 If $g_2$ is blue, then the red-blue pairs $e,g_2$ and $g_1,h_1$ satisfy the theorem. Otherwise,
the red-blue pairs $g_1,e^{\prime}$ and $g_2,h_2$ satisfy the
theorem. By symmetry, We may assume that $\{v_{k+1}\}\cup W_2$ is
also blue.

\bigskip
Next we observe that the hyperedge $f_1=\{v_2\}\cup W_2$, is red.
If not, consider $w_1\in W_1, w_2\in W_2$ and set
$f^{\prime}=\{w_1,w_2\}\cup (W\setminus (W_1\cup W_2))$. But then
$e^{\prime},f_1,f^{\prime},g_1$ is a blue copy of
$\mathcal{C}^k_4$. By a similar argument, $f_2=\{v_3\}\cup W_1$
must be red. However, we may assume that $b=\{v_1\}\cup W_2$ is
also red, otherwise $f_1,b$ and $f_2,h_1$ are two disjoint
red-blue pairs. Now the red-blue pairs $b,h_2$ and
$f_2,e^{\prime}$ are disjoint and satisfying the theorem. This
observation completes the proof. $\hfill\square$

\medskip
\begin{corollary}\label{C4-C4}
$R(\mathcal{C}^3_4,\mathcal{C}^3_4)=9$.
\end{corollary}
\noindent\textbf{Proof. }Suppose on the contrary that
$\mathcal{K}^3_9$ is 2-edge colored red and blue without a
monochromatic $\mathcal{C}^3_4$. By Lemma \ref{partition
Property}, we have two pairs of red-blue hyperedges $e,e^{\prime}$
and $f,f^{\prime}$ such that $(e\cup e^{\prime})\cap (f\cup
f^{\prime})=\emptyset$. Using Remark \ref{start} for $e\cup
e^{\prime}$ and $f\cup f^{\prime}$, we may assume that $|e\cap
e^{\prime}|=|f\cap f^{\prime}|=2$. Let $e=\{v_1,v_2,v_3\}$,
$e^{\prime}=\{v_2,v_3,v_4\}$ and $f=\{w_1,w_2,w_3\}$,
$f^{\prime}=\{w_2,w_3,w_4\}$. There is a single vertex $u$
uncovered by these four edges. W.l.o.g let $\{v_2,u,w_2\}$ be red,
then $g=\{v_1,v_4,w_3\}$ and $h=\{w_1,w_4,v_3\}$ are both blue,
otherwise we have a red $\mathcal{C}^3_4$. Now,
$e^{\prime},g,f^{\prime},h$ form a blue $\mathcal{C}^3_4$, a
contradiction. $\hfill \square$
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\bigskip
\begin{theorem}\label{Ramsey number of C4}
For every $k\geq 3$, $R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$.
\end{theorem}
\noindent\textbf{Proof. }The case $k=3$ follows from Corollary
\ref{C4-C4}, so we may assume that $k\ge 4$. Suppose indirectly
that the edges of $\mathcal{K}_{4k-3}^k$ can be colored red and
blue without monochromatic $\mathcal{C}^k_4$. Clearly,
$\mathcal{F}_{red}$ and $\mathcal{F}_{blue}$ are both nonempty. By
Remark \ref{start} and Lemma \ref{partition Property}, we can
select $f=\{w_1,w_2,\ldots, w_k\}\in \mathcal{F}_{red}$,
$f^{\prime}=\{w_2,w_3,\ldots, w_{k+1}\}\in \mathcal{F}_{blue}$ and
$e=\{v_1,v_2,\ldots, v_k\}\in \mathcal{F}_{red}$,
$e^{\prime}=\{v_2,v_3,\ldots, v_{k+1}\}\in \mathcal{F}_{blue}$ so
that $(f\cup f^{\prime})\cap (e\cup e^{\prime})=\emptyset$. Set
$V=V(\mathcal{K}_{4k-3}^k)$ and let $W=V\setminus (e\cup
e^{\prime}\cup f\cup f^{\prime})$, note that $|W|=2k-5$. Partition
the vertices of $W$ into two sets $A$ and $B$ with $|A|=|B|=k-3$
and let $u$ be the remaining vertex of $W$. In the following
claims, we let $T\subseteq A$, $S\subseteq B$ such that $|T|=t\ge
1$, $|S|=s$ and $s+t=k-3$. Also let $t_1\in T$ and $s_1\in
B\setminus S$ be arbitrary. Set $T^{\prime}=A\setminus T$,
$S^{\prime}=B\setminus S$, note that $|T^{\prime}|=s$ and
$|S^{\prime}|=t$.

\bigskip
\noindent \textbf{Claim 1. }If $i\in\{2,3,\ldots,k\}$ and
$e_1=\{w_1,w_{k+1},v_i\}\cup T\cup S$
 is red, then for any $j\in\{2,3,\ldots,k\}$, the hyperedge $g_1=\{v_1,v_{k+1},w_j\}\cup (T\setminus t_1)\cup (S\cup \{s_1\})$
 is also red.


\bigskip \noindent{\it Proof. }Let $m\in\{2,3,\ldots,k\}\setminus \{i\}$.
 The hyperedge
$f_1=\{u,v_m,w_{j}\}\cup T^{\prime}\cup S^{\prime}$ is blue,
otherwise $e,f_1,f,e_1$ form a red $\mathcal{C}^k_4$. By a similar
argument, for $i\ne j$, $h=\{v_1,v_{k+1},w_i\}\cup T\cup S$ is red
(otherwise $e^{\prime},f^{\prime},f_1,h$ is blue
$\mathcal{C}^k_4$) and $f_2=\{u,w_1,w_{k+1},v_i\}\cup
T^{\prime}\cup(S^{\prime}\setminus s_1)$ is blue (otherwise
$e,f,h,f_2$ is red $\mathcal{C}^k_4$). Now, if $g_1$ is blue then
$e^{\prime},g_1,f^{\prime},f_2$  is a blue $\mathcal{C}^k_4$, a
contradiction, thus $g_1$ is red as claimed.

\bigskip By symmetry, we have the following claim, which is given without proof.


\bigskip
\noindent \textbf{Claim 2. }If   $i\in \{2,3,\ldots,k\}$ and
$e_1=\{v_1,v_{k+1},w_i\}\cup T\cup S$ is red, then for any
$j\in\{2,3,\ldots,k\}$, the hyperedge $g_1=\{w_1,w_{k+1},v_j\}\cup
(T\setminus t_1)\cup (S\cup \{s_1\})$ is also red.

\bigskip

To finish the proof, we need the following.

\bigskip
\noindent \textbf{Claim 3. }If $e_1=\{v_2,w_{2},u\}\cup T\cup S$
is red, then $g_1=\{w_1,w_{k+1},v_i\}\cup (T\setminus t_1)\cup
(S\cup \{s_1\})$ is red for any $i\in\{3,4,\ldots,k\}$.

\bigskip \noindent{\it Proof. }Let $i\in\{3,4\ldots,k\}$.
The hyperedge $f_1=\{v_1,v_{k+1},w_j\}\cup T^{\prime}\cup
S^{\prime}$ is blue for for any $j\in\{3,4,\ldots,k\}$, otherwise
$e,f_1,f,e_1$ would form a red $\mathcal{C}^k_4$. Then
$h=\{w_1,w_{k+1},v_i\}\cup T\cup S$ is red otherwise
$e^{\prime},f_1,f^{\prime},h$ would give a blue $\mathcal{C}^k_4$.
The edge $f_2=\{v_1,v_{k+1},w_j,u\}\cup T^{\prime}\cup
(S^{\prime}\setminus s_1)$ is blue otherwise
$e^{\prime},f_2,f^{\prime},h$ is a blue $\mathcal{C}^k_4$.   Now,
if $g_1$ is blue then $e^{\prime},g_1,f^{\prime},f_2$  is a blue
$\mathcal{C}^k_4$, a contradiction. Therefore $g_1$ is red as
claimed.

\bigskip
 Now we finish the proof of the theorem. Without loss of generality, let $e_1=\{v_2,w_2,u\}\cup A$ be red. Using Claim 3,
 $\{w_1,w_{k+1},v_i\}\cup(T\setminus t_1)\cup (S\cup \{s_1\})$, $i\in\{3,4,\ldots,k\}$, is red and so using Claim 1 and Claim 2
  successively
$k-3$ times and switching the pairs $(v_3,w_3),(v_4,w_4)$
(starting with $t=k-3$), we obtain  that the edge
$h_1=\{v_1,v_{k+1},w_3\}\cup B$ is red,  a contradiction, since
$e,e_1,f,h_1$ is a red $\mathcal{C}^k_4$.
 This observation completes the proof.
\phantom{spacer}$\hfill \square$
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\vspace{.5 cm} From Lemma \ref{No Cn} and Theorem \ref{Ramsey
number of C4} we get the following.

\bigskip
\begin{corollary}\label{P_4}
For every $k\geq 3$,
$R(\mathcal{P}^k_4,\mathcal{P}^k_4)=R(\mathcal{C}^k_4,\mathcal{P}^k_4)=4k-2$.
\end{corollary}

\bigskip Proving
$R(\mathcal{C}^k_3,\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-1$ for
$k\ge3$ seems difficult, we could do it only for $k=3$.

\medskip

\begin{theorem}\label{C3-C3-C3}
$R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8$.
\end{theorem}
\noindent\textbf{Proof. } We suppose that the third color is green
and the vertex set is $V$. The lower bound is obtained by coloring
triples of $\{1,2,\dots 7\}$ by red if the smallest element in the
triple is 1, blue if the smallest element in the triple is 2 and
green otherwise. To see the upper bound, we consider some cases.

\bigskip
\noindent\textbf{Case 1. }There is a monochromatic, say green
$S=\mathcal{S}^3_3$ with edges $e_1=\{A,1,4\}, e_2=\{A,2,5\}$ and
$e_3=\{A,3,6\}$.

\bigskip
There is one vertex, $B$, not covered by $S$. Set
$U=\{1,2,3\},W=\{4,5,6\}$ and let $H$ be the graph obtained from
the complete graph spanned by $U\cup W$ by removing the pairs
$14,25,36$. Clearly, for any edge $ij$ of $H$, the hyperedge
$\{B,i,j\}$ is red or blue, otherwise we have a green
$\mathcal{C}^3_3$. For any edge $ij$ of $H$ we define a \it mirror
edge $kl$ \rm so that $k$ is the third vertex in the edge of $S$
containing $\{A,i\}$ and $l$ is the third vertex of the edge of
$S$ containing $\{A,j\}$.

\bigskip
\noindent\textbf{Subcase 1.1. }For any edge $ij$ of $H$ and its
mirror edge $kl$, the hyperedges $\{i,j,B\},\{k,l,B\}$ have
different colors (one of them red and the other is blue).

\bigskip
Among the hyperedges $\{1,5,B\},\{2,6,B\},\{3,4,B\}$ two colored
with the same color, say the first two is colored with red. By the
subcase assumption, the hyperedges $\{2,4,B\},\{3,5,B\}$ are blue.
Now the hyperedge $f=\{4,5,6\}$ defines a monochromatic
$\mathcal{C}^3_3$: if green, with $e_1,e_2$; if red, with
$\{1,5,B\},\{2,6,B\}$; if blue, with $\{2,4,B\},\{3,5,B\}$.

\bigskip
\noindent\textbf{Subcase 1.2. }There exists an edge of $H$, say
$15$, such that with its mirror edge $24$, the hyperedges
$\{1,5,B\},\{2,4,B\}$ are both red.

\bigskip
We claim that for any vertex $v\in V\setminus \{1,2,4,5\}$ (there
are four such vertices) the hyperedges $\{1,5,v\},\{2,4,v\}$ are
red, the hyperedges $\{1,4,v\}, \{2,5,v\}$ are green and the
hyperedges $\{1,2,v\},\{4,5,v\}$ are blue. We call such a
quadruple $Q=\{1,2,4,5\}$ a {\it factored quadruple}. First
observe that the hyperedges $\{1,2,3\},\{1,2,6\},\{4,5,3\}$ and
$\{4,5,6\}$ are blue, otherwise we have a monochromatic
$\mathcal{C}^3_3$. Similarly, the hyperedges $\{1,4,6\},\{2,5,6\}$
are green. These observations with the initial assumptions are
easily imply that  $Q$ is factored from $A$ as required. An easy
consequence is that $Q$ is factored from the other three vertices
in the same way.

\bigskip
At this point we introduce new notations, consider a factored
quadruple $Q=\{A_1,A_2,A_3,A_4\}$ and its complement $R=V\setminus
Q=\{1,2,3,4\}$. Two of the four hyperedges in $R$ have the same
color, say $\{1,2,3\},\{2,3,4\}$ are green. Let $G_R$ be the graph
with vertex set $R$ and with all edges except $23$. Observe that
for any edge $ij\in E(G_R)$, the hyperedge $f=\{A_k,i,j\}$ is red
or blue. Indeed, otherwise consider the green hyperedge $e$ in $R$
that splits $ij$, say $i\in e, j\notin e$. Since $Q$ is factored,
there is a green hyperedge $g$ with one vertex in $e\setminus
\{i\}$ that intersects $Q$ in two vertices and $A_k\in g$. The
edges $e,f,g$ would form a green $\mathcal{C}^3_3$.

\bigskip
Now we may assume that $e=\{1,4,A_1\}$ is red. Since $Q$ is
factored, there is $A_i\ne A_1$, say $A_2$, such that
$\{j,A_1,A_2\}$ is red for all $j\in R$. If $f=\{1,3,A_2\}$ is
red, then $e,f,\{2,A_1,A_2\}$ would form a red $\mathcal{C}^3_3$.
Thus $f$, and similarly $g=\{2,4,A_2\}$ is blue. Since $Q$ is
factored, with one of $A_3,A_4$, say with $A_3$, $\{j,A_2,A_3\}$
is blue for all $j\in R$. Now $\{1,2,A_3\}$ must be red, otherwise
$\{1,2,A_3\},f=\{1,3,A_2\},\{4,A_2,A_3\}$ would form a blue
$\mathcal{C}^3_3$. Similarly $\{3,4,A_3\}$ must be red and so
hyperedges $\{1,2,A_3\},\{3,4,A_3\},\{1,4,A_1\}$ form a  red
$\mathcal{C}^3_3$.

\noindent\textbf{Case 2. } There is no monochromatic $S=\mathcal{S}^3_3$.

\bigskip
It is easy to see (also follows from a well-known result
\cite{CL}) that (centered at any vertex $A$) there is a
monochromatic, say green,  $T=\mathcal{S}^3_2$, with edges
$\{A,1,2\},\{A,3,4\}$. Let $U=\{5,6,7\}$ denote the set of
vertices uncovered by $T$.

\bigskip
\noindent\textbf{Subcase 2.1. } There is a monochromatic (red or
blue) $\mathcal{S}^3_2$ in the form $\{i,1,3\},\{i,2,4\}$ or
$\{i,1,4\},\{i,2,3\}$, $i\in U$.

\bigskip
Suppose w.l.o.g that the hyperedges $\{i,1,3\},\{i,2,4\}$ are red.
Then  $\{j,1,4\},\{j,2,3\}$ must be blue for $j\in U, j\ne i$,
otherwise we have green or red $\mathcal{C}^3_3$. Then no matter
what is the color of the edge $\{A,i,j\}$, we have a monochromatic
$\mathcal{S}^3_3$ and this leads back to the Case 1.

\bigskip
\noindent\textbf{Subcase 2.2. } For every $i\in U$ the pairs of
edges $\{i,1,3\},\{i,2,4\}$ and $\{i,1,4\},\{i,2,3\}$ have
different (red-blue) colors.

\bigskip
From the subcase definition w.l.o.g. we find two edges
$\{1,3,i\},\{2,3,j\}$ of the same color, say red, such that
$i,j\in U, i\ne j$. This implies that $\{A,i,j\}$ is blue (green
would give green $\mathcal{S}^3_3$, red would give red
$\mathcal{C}^3_3$). The edge $\{2,4,i\}$ is also blue (green or
red both would give a monochromatic $\mathcal{C}^3_3$). No matter
what is the color of $\{1,4,j\}$, a monochromatic
$\mathcal{C}^3_3$ arises. $\hfill \square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip
\section{\large Concluding remarks}

\medskip
\hspace{.5 cm}It would be interesting to decide whether the
natural lower bound of Lemma \ref{lower bound} is always the true
value of the Ramsey numbers for $k$-uniform (loose) paths and
cycles if $k\ge 3$. The test case is $k=3$.

\medskip
\begin{question}
For every $n\geq m\geq 2$,
$R(\mathcal{P}_n^3,\mathcal{P}_m^3)=R(\mathcal{P}_n^3,\mathcal{C}_m^3)=
R(\mathcal{C}_n^3,\mathcal{C}_m^3)+1=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor?$
In particular,
$$R(\mathcal{P}_n^3,\mathcal{P}_n^3)=R(\mathcal{C}_n^3,\mathcal{C}_n^3)+1=\Big\lceil\frac{5n}{2}\Big\rceil?$$
\end{question}

\medskip
We answer this question affirmatively when one of $m,n$ is small,
at most $6$, in a paper under preparation \cite{prep}.


\bigskip
In certain cases one could get estimates of the Ramsey numbers
addressed in this paper by applying Tur\'an-type results. For
example, the behavior of the $r$-color Ramsey number of triangles
in hypergraphs is essentially different from the graph case. While
$R(C_3,C_3,\dots,C_3)$ is at least exponential in $r$ (for $r$
arguments), it is linear for hypergraphs.

\medskip
\begin{proposition}\label{rcolor}For $r$ arguments, $r+5\le
R(\mathcal{C}_3^3,\mathcal{C}_3^3,\dots, \mathcal{C}_3^3)\le
3r+1.$
\end{proposition}

\noindent\textbf{Proof. }The lower bound is obtained by coloring
triples of $\{1,2,\dots r+4\}$ by their smallest element $i$ if
$i<r$, otherwise by color $r$. The upper bound follows from the
Tur\'an number of $\mathcal{C}_3^3$ (see \cite{FK}): If $n\geq 6$,
$\mathcal{F}\subseteq {n \choose 3}$ and $\mathcal{F}$ contains no
$\mathcal{C}_3^3$, then $|\mathcal{F}|\leq {n-1\choose 2}$.
Therefore, for the smallest $n$ for which
$${n \choose 3}> r {n-1\choose 2},$$
every $r$-colored $\mathcal{K}_n^3$ will contain a monochromatic
$\mathcal{C}_3^3$, implying the upper bound. $\hfill \square$

\vspace{.7 cm} It is known from \cite{FK} that if $n\geq 8$,
$\mathcal{F}\subseteq {n \choose 3}$, $|\mathcal{F}|={n-1 \choose
2}$ and $\mathcal{F}$ contains no $\mathcal{C}_3^3$, then all
elements of $\mathcal{F}$ goes through a fix point. Using this
fact, the upper bound of the previous proposition can be reduced
to $3r$ for $r\geq3$. We proved here that for $r=2,3$ the lower
bound is the truth and we suspect that this is true for all $r$.

\bigskip
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%\footnotesize

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% we recommend using BibTeX to create a bibliography
% \bibliographystyle{plain}
% \bibliography{myBibFile}
% then copy and past the contents of your .bbl file into this .tex file

\begin{thebibliography}{10}

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stripes, {\em J. Austral. Math. Soc.} {\bf 19} (1975),
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\bibitem{FK} R. Cs\'ak\'any, J. Kahn, A homological approach to two
problems on finite sets, {\em J. Algebraic Combin.} {\bf 9}
(1999), 141-149.\vspace{.1 cm}


\bibitem{Ramsy number of cycles}
R.J. Faudree, S.L. Lawrence, T.D. Parsons,  R.H. Schelp,
Path-Cycle Ramsey Numbers, {\it Discrete Math.} {\bf 10} (1974),
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\bibitem{Ramsey number of cycle-path}
R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in
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cm}

\bibitem{Ramsey number of paths}
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cm}

\bibitem{GSS}
A. Gy\'{a}rf\'{a}s, G. S\'{a}rk\"{o}zy, E. Szemer\'{e}di, The
Ramsey number of diamond-matchings and loose cycles in
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\#R126.\vspace{.1 cm}

\bibitem{prep} A. Gy\'{a}rf\'{a}s, G. Raeisi, Some Ramsey numbers for loose cycles and paths in 3-uniform
hypergraphs, in preparation.\vspace{.1 cm}

\bibitem{HLR}
P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. Ruci\'{n}ski, M.
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%\bibitem{Bollobas} B{\'e}la Bollob{\'a}s.  \newblock Almost every
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\end{thebibliography}

\end{document}
