\documentclass[12pt]{article}
\usepackage{amsfonts,amssymb,amsthm, amsmath}
\usepackage{e-jc}
\specs{P12}{19}{2012}

\newcommand{\w}{\omega}
\newcommand{\IN}{\mathbb N}
\newcommand{\IQ}{\mathbb Q}
\newcommand{\e}{\varepsilon}
\newcommand{\Iso}{\mathrm{Iso}}
\newcommand{\Aff}{\mathrm{Aff}}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\diam}{\mathrm{diam}}
\newcommand{\IR}{\mathbb R}
\newcommand{\IZ}{\mathbb Z}
\newcommand{\A}{\mathcal A}
\newcommand{\F}{\mathfrak{F}}
                                                                                                                                                                       \newcommand{\St}{\mathcal{S}t}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}
\newtheorem{example}[theorem]{Example}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\title{Kaleidoscopical configurations in $G$-spaces}

\author{Taras Banakh\\
\small Faculty of Mechanics and Mathematics\\[-0.8ex]
\small Lviv University, Universytetska 1, Lviv 79000, Ukraine\\
\small \texttt{t.o.banakh@gmail.com}\\
\and
Oleksandr Petrenko\\
\small Faculty of Cybernetics\\[-0.8ex]
\small Kyiv University, Volodymyrska 64, Kyiv 01033, Ukraine\\
\small \texttt{opetrenko72@gmail.com}\\
\and
Igor Protasov\\
\small Faculty of Cybernetics\\[-0.8ex]
\small Kyiv University, Volodymyrska 64, Kyiv 01033, Ukraine\\
\small \texttt{i.v.protasov@gmail.com}\\
Sergiy Slobodianiuk\\
\small Faculty of Mechanics and Mathematics\\[-0.8ex]
\small Kyiv University, Volodymyrska 64, Kyiv 01033, Ukraine\\
\small \texttt{slobodianiuk@yandex.ru}\\
}

\date{\dateline{Jan 27, 2011}{Dec 13, 2011}{Jan 6, 2012}\\
\small Mathematics Subject Classifications: 05B45, 05C15, 05E15, 05E18, 20K01}

\begin{document}
\maketitle

\begin{abstract}
Let $G$ be a group and $X$ be a $G$-space with the action $G\times
X\rightarrow X$, $(g,x)\mapsto gx$. A subset $F$ of $X$ is called a
{\em kaleidoscopical configuration} if there exists a coloring
$\chi:X\rightarrow C$ such that the restriction of $\chi$ on each
subset $gF$, $g\in G$, is a bijection. We present a construction
(called the splitting construction) of kaleidoscopical
configurations in an arbitrary $G$-space, reduce the problem of
characterization of kaleidoscopical configurations in a finite
Abelian group $G$ to a factorization of $G$ into two subsets, and
describe all kaleidoscopical configurations in isometrically
homogeneous ultrametric spaces with finite distance scale. Also we
construct $2^{\mathfrak c}$ (unsplittable) kaleidoscopical
configurations of cardinality $\mathfrak c$ in the Euclidean space
$\IR^n$.
\end{abstract}

\section*{Introduction}
Let $X$ be a set and $\mathfrak{F}$ be a family of subsets of $X$
(the pair $(X,\mathfrak{F})$ is called a {\em hypergraph}).
Following \cite{BP}, we say that a coloring $\chi:X\to\kappa$ of $X$
(i.e. a mapping of $X$ onto a cardinal $\kappa$) is
\begin{itemize}
\item $\mathfrak{F}$-{\em surjective} if the restriction $\chi|_F$ is surjective for all $F\in\mathfrak{F}$;
\item $\mathfrak{F}$-{\em injective} if $\chi|_F$ is injective for all $F\in\mathfrak{F}$;
\item $\mathfrak{F}$-{\em bijective} or $\mathfrak{F}$-{\em kaleidoscopical} if $\chi|_F$ is bijective for all $F\in\mathfrak{F}$.
\end{itemize}

A hypergraph $(X,\mathfrak{F})$ is called {\em kaleidoscopical} if
there exists an $\mathfrak{F}$-kaleidoscopical coloring
$\chi:X\to\kappa$. The adjective ``kaleidoscopical" appeared in
definition \cite{PP} of an $s$-regular graph $\Gamma(V,E)$ (each vertex
$v\in V$ has degree $s$) admitting a vertex $(s+1)$-colloring such
that each unit ball $B(v,1)=\{u\in V:d(u,v)=1\}$ has the vertices of
all colors ($d$ is a path metric on $V$). These graphs can be
considered as a graph counterpart of Hamming codes \cite{PO}.

We shall consider hypergraphs related to a $G$-space. Let $G$ be a
group. A $G$-{\em space} is a set $X$ endowed with an action
$G\times X\to X$, $(g,x)\mapsto gx$. All $G$-spaces are suppose to
be transitive (for any $x,y\in X$ there exists $g\in G$ such that
$gx=y$). For a subset $A\subseteq X$, we put $G[A]=\{gA:g\in G\}$.

A subset $A\subseteq X$ is called a {\em kaleidoscopical
configuration} if the hypergraph $(X,G[A])$ is kaleidoscopical (in
words, if there exists a coloring $\chi:X\to |A|$ such that
$\chi|_{gA}$ is bijective for every $g\in G$).

In Section~\ref{s:groups} we show that kaleidoscopical
configurations are tightly connected with classical combinatorial
theme {\em Transversality} and, in the case $X=G$ and (left) regular
action of $G$ on $G$, with factorization problem, well known in {\em
Factorization Theory of Groups}, see \cite{Sz}, \cite{SS}.

In Section~\ref{s:split} we introduce and describe the
kaleidoscopical configurations (called splittable) which arise from
the chains of $G$-invariant equivalences (imprimitivities) on $X$.
If a $G$-space $X$ is primitive (the only $G$-invariant equivalences
on $X$ are $X\times X$ and $\Delta_X$) then the splittable
configurations in $X$ are only $X$ and the singletons.

In Section~\ref{s:prim} we prove that every kaleidoscopical
configuration in an isometrically homogeneous metric space with finite
distance scale is splittable. For $n\geq 2$, we construct a plenty
of kaleidoscopical configurations of cardinality $\mathfrak{c}$ in
$\IR^n$. These configurations are non-splittable because $\IR^n$ is
isometrically primitive. We don't know whether there exists a finite
non-singleton or countable kaleidoscopical configurations in
$\IR^n$, $n\geq 2$.

In Section~\ref{s:Hajos} we  study the problem of splittability of
kaleidoscopical configurations in finite Abelian groups and
reformulate this problem in terms of the semi-Haj\'os property, which is a weak version of the Haj\'os property well-known in the factorization theory of groups \cite{Sz}, \cite{SS}.

We note also that kaleidoscopical configurations in a sense are
antipodal to mono\-chromatizable configurations defined and studied in
\cite[Chapter 8]{BP}: a subset $A$ of a $G$-space $X$ is called {\em
monochromatizable} if, for any finite coloring of $X$, there exists
$g\in G$ such that $gA$ is monochrome.

%For more information on kaleidoscopical configurations in graphs and hypergraphs, see \cite{BP}, \cite{PP} and \cite{PO}.

\section{Transversality and factorization}\label{s:groups}

Let $(X,\mathfrak{F})$ be a hypergraph. A subset $T\subseteq X$ is
called an {\em $\mathfrak{F}$-transversal} if $|F\bigcap T|=1$ for each
$F\in\mathfrak{F}$.

\begin{proposition} A hypergraph $(X,\mathfrak{F})$ is
kaleidoscopical if and only if $X$ can be partitioned into
$\mathfrak{F}$-transversals.
\end{proposition}
\begin{proof}
For a kaleidoscopic hypergraph $(X,\mathfrak{F})$, let
$\chi:X\to\kappa$ be a kaleidoscopical coloring. Then
$X=\bigsqcup_{\alpha<\kappa}\chi^{-1}(\alpha)$ is a partition of $X$
into $\mathfrak{F}$-transversal.

On the other hand, if $X=\bigsqcup_{\alpha<\kappa}T_\alpha$ is a
partition of $X$ into $\mathfrak{F}$-transversal subsets, then the coloring
$\chi:X\to\kappa$ defined as $\chi(x)=\alpha\Leftrightarrow x\in
T_\alpha$ is kaleidoscopical.
\end{proof}

Let $X$ be a $G$-space, $A$ be a kaleidoscopical configuration in
$X$. If $T$ is $G[A]$-transversal then $A$ is $G[T]$-transversal and
$gT$ is $G[A]$ transversal for each $g\in G$.

We say  that a kaleidoscopical configuration $A$ in $X$ is {\em
homogeneous} if there exist a $G[A]$-transversal $T$ and a subset
$H\subseteq X$ such that $X=\bigsqcup_{h\in H}hT$.

A subset $A$ of a group $G$ is defined to be {\em complemented} in
$G$ if there exists a subset $B\subseteq G$ such that the
multiplication mapping $\mu:A\times B\to G$, $(a,b)\mapsto ab$, is
bijective. Following \cite{SS}, we call the set $B$ a {\em
complementer factor} to $A$, and say that $G=AB$ is a {\em
factorization} of $G$. In this case, we have the partitions 
$$G=\bigsqcup_{a\in A}aB=\bigsqcup_{b\in B}Ab.$$
A subset $A\subseteq G$ is called {\em doubly complemented} if there
are factorization $G=AB=BC$ for some subsets $B,C$ of $G$.
\begin{proposition}\label{factor} For two subsets $A,B$ of a group $G$ the following conditions are equivalent:
\begin{enumerate}
\item $B$ is $G[A]$-transversal;
\item $G=AB^{-1}$ is a factorization of $G$.
\end{enumerate}
\end{proposition}

\begin{proof} $(1)\Ra(2)$ For each $g\in G$, $g^{-1}A\cap B\ne\emptyset$, so $g\in AB^{-1}$. If $g=a_1b_1^{-1}=a_2b_2^{-1}$ for some $a_1,a_2\in A$, $b_1,b_2\in B$, then $g^{-1}a_1=b_1$ and $g^{-1}a_2^{-1}=b_2$ and by (1), $b_1=b_2$ and $a_1=a_2$, witnessing that $G=A B^{-1}$ is a factorization of $G$.
\smallskip

$(2)\Ra(1)$ Fix any $g\in G$. The inclusion $g^{-1}\in AB^{-1}$ implies $gA\cap B\ne\emptyset$. If $ga_1=b_1$ and $ga_2=b_2$ for some $a_1,a_2\in A$, $b_1,b_2\in B$, then $g^{-1}=a_1b_1^{-1}=a_2b_2^{-1}$ and by (2), $b_1=b_2$, witnessing that $|gA\cap B|=1$.
\end{proof}

\begin{corollary} Each kaleidoscopical configuration in a group $G$
is complemented.
\end{corollary}

\begin{proof} Given a kaleidoscopical configuration $A\subset G$, fix an $A$-kaleidoscopical coloring $\chi:G\to C$. We choose a color $c\in C$, consider the monochrome class $B=\chi^{-1}(b)\subset G$ and observe that for every $g\in G$ \ $|gA\cap B|=1$ by the definition of $A$-kaleidoscopical coloring. By Proposition~\ref{factor}, $G=AB^{-1}$ is a factorization, so $A$ is complemented in $G$.
\end{proof}

\begin{proposition}\label{p1.4} A subset $A$ of a group $G$ is doubly
complemented if and only if $A$ is a homogeneous kaleidoscopical
configuration.
\end{proposition}

\begin{proof} Let $G=AB=BC$ be factorizations of $G$. By
Proposition~\ref{factor}, $B^{-1}$ is a $G[A]$-transversal. Since
$G=\bigsqcup_{c\in C}c^{-1}B^{-1}$, we conclude that $A$ is a homogeneous
kaleidoscopical configuration.

Let $A$ be a homogeneous kaleidoscopical configuration. We choose a
$G[A]$-transversal $T$ and a subset $H\subseteq G$ such that
$G=\bigsqcup_{h\in H}hT$. By proposition~\ref{factor}, $G=AT^{-1}$.
Since $G=\bigsqcup_{h\in H}hT$, $G=T^{-1}H^{-1}$ is a factorization.
Hence, $A$ is doubly complemented.
\end{proof}

\begin{corollary}\label{c1.5} For a subset $A$ of an Abelian group $G$, the
following statements are equivalent:
\begin{enumerate}
\item $A$ is complemented;
\item $A$ is a kaleidoscopical configuration;
\item  $A$ is a homogeneous kaleidoscopical configuration.
\end{enumerate}
\end{corollary}

\begin{question} Is each complemented subset of a (finite) group
kaleidoscopical?
\end{question}

\begin{proposition}\label{section} Let $X$ be a transitive $G$-space, $x\in X$, $G_x=\{g\in
G:gx=x\}$, $\gamma_x: G\to X$, $\gamma_x(g)=gx$, $s:X\to G$ be a
section of $\gamma_x$. Let $A$ be a subset of $X$, $T$ be a
$G[A]$-transversal. Then
\begin{enumerate}
\item $s(T)$ is a $G[\gamma_x^{-1}(A)]$-transversal;
\item $|G|=|G_x||A||T|$.
\end{enumerate}
\end{proposition}

\begin{proof} The statement $(1)$ is evident. The statement $(2)$
follows from $(1)$ and Proposition~\ref{factor}.
\end{proof}

\begin{corollary}\label{c1.8} Let $A$ be a kaleidoscopical configuration in a
finite transitive $G$-space $X$ with a kaleidoscopical coloring  $\chi:G\to k$.
Then $\chi^{-1}(0)=\dots=\chi^{-1}(k-1)$ and
$|X|=|A||\chi^{-1}(0)|$.
\end{corollary}

\begin{proof} We may suppose that $G$ is a subgroup of the group of
all permutations of $X$ so $G$ is finite. Since $|G|=|X||G_x|$, we
can apply Proposition~\ref{section}(2).
\end{proof}

\begin{proposition} Let $\kappa$ be an infinite cardinal, $(X,\mathfrak{F})$
be a hypergraph such that $|\mathfrak{F}|=\kappa$ and $|F|=\kappa$
for each $F\in\mathfrak{F}$. If $|F\cap F'|<cf\kappa$ for all
distinct $F,F'\in\mathfrak{F}$ then there is a disjoint family
$\mathfrak{T}$ of $\mathfrak{F}$-transversals such that
$|\mathfrak{T}|=\kappa$ and $|T|=\kappa$ for each $T\in\mathfrak{T}$
\end{proposition}
\begin{proof} We enumerate $\mathfrak{F}=\{F_\alpha:\alpha<\kappa\}$
and choose inductively the subsets $\{V_\alpha\subset
F_\alpha:\alpha<\kappa\}$ such that the family $\{F_\alpha\setminus
V_\alpha:\alpha<\kappa\}$ is disjoint and $|F_\alpha\setminus
V_\alpha|=\kappa$ for each $\alpha<\kappa$. Let $F_\alpha\setminus
V_\alpha=\{t_{\alpha\beta}:\beta<\kappa\}$,
$T_\beta=\{t_{\alpha\beta}:\alpha<\kappa\}$. Then
$\mathfrak{T}=\{T_\beta:\beta<\kappa\}$ is the desired family.
\end{proof}
For a hypergraph $(X,\mathfrak{F})$, $x\in X$ and $A\subseteq X$, we
put
$$St(x,\mathfrak{F})=\bigcup\{F\in\mathfrak{F}:x\in F\},$$
$$St(A,\mathfrak{F})=\bigcup\{St(a,F):a\in A\}.$$
\begin{proposition}\label{p1.10} A hypergraph $(X,\mathfrak{F})$ is
kaleidoscopical provided that, for some infinite cardinal $\kappa$,
the following two conditions are satisfied:
\begin{enumerate}
\item $|\mathfrak{F}|\leq\kappa$ and $|F|=\kappa$ for each $F\in\mathfrak{F}$;
\item for any subfamily $\mathfrak{A}\subset\mathfrak{F}$ of cardinality $|\mathfrak{A}|<\kappa$ and any subset $B\subset X\setminus(\bigcup\mathfrak{A})$ of cardinality $|B|<\kappa$ the intersection $St(B,\mathfrak{F})\cap(\bigcup{\mathfrak{A}})$ has cardinality less than $\kappa$.
\end{enumerate}
\end{proposition}
\begin{proof} Let $\lambda=|\F|$ and $\F=\{F_\alpha:\alpha<\lambda\}$ be an injective enumeration of $\F$. By induction we shall construct a transfinite sequence $(\chi_\alpha:F_\alpha\to\kappa)_{\alpha<\lambda}$ of bijective colorings such that for any ordinals $\alpha<\beta<\lambda$
\begin{enumerate}
\item[$(1_{\alpha\beta})$] the colorings $\chi_\alpha$ and $\chi_\beta$ coincide on $F_\alpha\cap F_\beta$;
\item[$(2_{\alpha\beta})$] no distinct points $a\in F_\alpha$ and $b\in F_\beta$ with $\chi_\alpha(a)=\chi_\beta(b)$ lie in some hyperedge $F\in\F$.
\end{enumerate}

Assume that for some ordinal $\gamma<\lambda$ we have constructed a
sequence of colorings $(\chi_{\alpha})_{\alpha<\gamma}$ satisfying
the conditions $(1_{\alpha\beta})$ and $(2_{\alpha\beta})$ for all
$\alpha<\beta<\gamma$.

Let us define a bijective coloring $\chi_\gamma:F_\gamma\to\kappa$.
First we show that the union
$$F_\gamma'=\bigcup_{\alpha<\gamma}F_\gamma\cap F_\alpha$$ has
cardinality $|F'_\gamma|<\kappa$. Observe that for each
$\alpha<\gamma$ we get $F_\alpha\not\subset F_\gamma$. Assuming
conversely that $F_\alpha\subsetneq F_\gamma$ and taking any point
$v\in F_\gamma\setminus F_\alpha$ we conclude that the intersection
$F_\alpha\cap\St(v,\F)\supset F_\alpha\cap F_\gamma=F_\alpha$ has
cardinality $\ge\kappa$, which contradicts the condition (2) of the
theorem.

Therefore, for each $\alpha<\gamma$ we can choose a point
$v_\alpha\in F_\alpha\setminus F_\gamma$. Then for the set
$B=\{v_\alpha:\alpha<\gamma\}$ the set $F_\gamma'\subset
F_\gamma\cap \St(B,\F)$ has cardinality
$|F_\gamma'|\le|F_\gamma\cap\St(A,\F)|<\kappa$ according to (2).

For every point $x\in F_\gamma\setminus F_\gamma'$ and every ordinal
$\alpha<\gamma$, we consider the sets $\St(x,\F)\cap F_\alpha$ and
$C_\alpha(x)=\chi_\alpha(\St(x,\F)\cap F_\alpha)\subset\kappa$. The
condition (2) implies that the set
$C(x)=\bigcup_{\alpha<\gamma}C_\alpha(x)$ has cardinality
$|C(x)|<\kappa$.

Let $\prec$ be any well-ordering on the set $F_\gamma$ such that
$F_\gamma'$ coincides the initial segment $\{x\in F_\gamma:x<y\}$
for some point $y\in F_\gamma$. Consider the coloring
$\chi_\gamma:F_\gamma\to\kappa$ defined by
$\chi_\gamma(x)=\chi_\alpha(x)$ if $x\in F_\gamma\cap F_\alpha$ for
some $\alpha<\gamma$ and
$$\chi_\gamma(x)=\min\{\kappa\setminus(C(x)\cup\{\chi(y):y\prec
x\})\}$$ if $x\in F_\gamma\setminus F_\gamma'$.

Let us show that the coloring $\chi_\gamma:F_\gamma\to \kappa$ is
bijective. The injectivity of $\chi_\gamma$ follows from the
definition of $\chi_\gamma$ and the conditions $(2_{\alpha\beta})$,
$\alpha<\beta<\gamma$.

The surjectivity of $\chi_\gamma$ will follow as soon as we check
that for each color $c\in\kappa\setminus\chi_\gamma(F_\gamma')$ the
set $F_\gamma(c)=\{x\in F_\gamma\setminus F_\gamma':c\in C(x)\}$ has
cardinality $<\kappa$. Observe that $c\in C(x)$ if and only if there
is $\alpha<\gamma$ and a point $a\in F_\alpha\setminus F_\gamma$
such that $\chi_\alpha(a)=c$ and $x\in \St(a,\F)$. The set
$A_c=\bigcup_{\alpha<\gamma}\chi_\alpha^{-1}(c)\setminus F_\gamma$
has size $|A_c|\le\gamma<\kappa$, and by the condition (2), the set
$F_\gamma(c)\subset F_\gamma\cap\St(A_c,\F)$ has cardinality
$<\kappa$. This completes the proof of the bijectivity of the
coloring $\chi_\gamma$.

The conditions $(1_{\alpha\gamma})$ and $(2_{\alpha,\gamma})$ for
all $\alpha<\gamma$ follow from the definition of the coloring
$\chi_\gamma$. This completes the inductive step of the construction
of the sequence $(\chi_\alpha)_{\alpha<\lambda}$.

After completing the inductive construction, let $\chi:V\to\kappa$
be any coloring such that $\chi|_{F_\alpha}=\chi_\alpha$ for all
$\alpha<\lambda$. The conditions $(1_{\alpha\beta})$ guarantee that
the coloring $\chi$ is well-defined. The bijectivity of the
colorings $\chi_\alpha$, $\alpha<\lambda$, ensures the
kaleidoscopicity of the coloring $\chi$.
\end{proof}
We conclude this section with short discussion of possibilities of
transfering above notions and results to quasigroups.

We recall that a {\it quasigroup} is a set $X$ endowed with a binary
operation $*:X\times X\to X$ such that, for every $a,b\in X$, the
system of equations $a*x=b$, $y*a=b$ has a unique solution
$x=a\backslash b$, $y=b/a$ in $X$.

In an obvious way the notion of a kaleidoscopical configuration generalizes to quasigroup.

A subset $A$ of a quasigroup $X$ is called
\begin{itemize}
\item {\it kaleidoscopical} if there is a coloring $\chi:X\to C$ such that $\chi|_{x*A}:x*A\to C$ is bijective for all $x\in X$;
\item {\it complemented} if there is a subset $B\subset X$ such that the right division $\delta:B\times A\to X$, $\delta(b,a)=b/a$ is bijective;
\item {\it doubly complemented} if there exists a complemented subset  $B\subset X$ such that the multiplication $\mu:A\times B\to X$, $\mu(a,b)=a*b$,  is bijective;
\item {\it self-complemented} if the maps $\mu:A\times A\to X$, $\mu(x,y)=x*y$, and $\delta:A\times A\to X$, $\delta(x,y)=x/y$, are bijective.
\end{itemize}
It follows from the proof of Proposition~\ref{factor} that each
kaleidoscopical subset in a quasigroup is complemented. In contrast,
Proposition~\ref{p1.4} does not generalize to quasigroup.
\begin{example} There exists a quasigroup $X$ of order $|X|=9$ that contains a self-complemented subset $A\subset X$, which is not kaleidoscopical.
\end{example}

\begin{proof} It is well-known that finite quasigroups can be identified with
Latin squares, i.e., $n\times n$ matrices whose rows and columns are
permutations of the set $\{1,\dots,n\}$. For $r,s\le n$ an $(r\times
s)$-matrix $(x_{ij})$ is called a {\em partial Latin $(r\times
s)$-rectangle} if $x_{ij}\in\{1,2,\dots,n\}$ and $x_{lj}\ne
x_{ij}\ne x_{ik}$ for any $1\le i\ne l\le r$ and $1\le j\ne k\le s$.
By the result of Ryser \cite{Ryser}  (see also Lemma 1 in
\cite[p.214]{PB}) each partial latin $(r\times s)$-rectangle can be
completed to a Latin $(n\times n)$-square if and only if each number
$i\in\{1,\dots,n\}$ appears in the rectangle not less than $r+s-n$
times. This extension result allows us to find a quasigroups
operation on $X=\{1,\dots,9\}$ whose multiplication table has the
following first three columns:
\[
\begin{tabular}{c|ccc}
$*$&1&2&3\\
\hline
1&1&4&5\\
2&6&2&7\\
3&8&9&3\\
\hline
4&4&1&6\\
5&5&6&1\\
\hline
6&2&7&8\\
7&7&8&2\\
8&3&5&9\\
9&9&3&4\\
\hline
\end{tabular}
\]

Looking at this table we can see that the set $A=\{1,2,3\}$ is
self-complemented as $A*A=X=A/A$. Assuming that $A$ is
kaleidoscopical, find a coloring $\chi:X\to A$ such that $\chi|_{x*A}$
is bijective for each $x\in X$. Since $1*A=\{1,4,5\}$ and
$4*A=\{4,1,6\}$, the elements $5$ and $6$ have the same color, which
is not possible as $5*A=\{5,6,1\}$ and $\chi|_{5*A}$ is bijective.
\end{proof}


\section{Splitting}\label{s:split}

In this section we present a simple construction of kaleidoscopical configurations in arbitrary $G$-space, called the splitting construction. Kaleidoscopic subsets constructed in this way will be called splittable.

First we recall some definitions. A mapping $\varphi:X\to Y$ between
$G$-spaces is called {\em equivariant} if
$\varphi(gx)=g\,\varphi(x)$ for all $g\in G$ and  $x\in X$. It is
easy to see that each equivariant mapping between transitive
$G$-spaces is surjective and homogeneous.

A function $\varphi:X\to Y$ is defined to be {\em homogeneous} if it is $\kappa$-to-1 for some non-zero cardinal $\kappa$. The latter means that $|\varphi^{-1}(y)|=\kappa$ for all $y\in Y$.

\begin{proposition}\label{p2.1} Let $\kappa$ be a non-zero cardinal, $\pi:X\to Y$ be an $\kappa$-to-1 equivariant mapping between two $G$-spaces and $s:Y\to X$ be a section of $\varphi$. Let $K\subset Y$ be a kaleidoscopic subset and $\chi:Y\to C$ be an $K$-kaleidoscopical coloring. Then:
\begin{enumerate}
\item the preimage $\bar K=\pi^{-1}(K)$ is a kaleidoscopical configuration in $X$ with respect to any coloring $\bar \chi:X\to C\times \kappa$ such that for each $y\in Y$ the restriction $\bar\chi|_{\varphi^{-1}(y)}:\pi^{-1}(y)\to\{\chi(y)\}\times \kappa$ is bijective;
\item the image $\tilde K=s(K)$ is a kaleidoscopical configuration in $X$ with respect to the $\tilde K$-kaleidoscopical coloring $\tilde\chi=\chi\circ \pi:X\to C$.
\end{enumerate}
\end{proposition}

\begin{proof} 1. Given any element $g\in G$, we need to check that the restriction $\bar \chi|_g\bar K:g\bar K\to C\times\kappa$ is bijective. To see that it is surjective, take any color $(c,\alpha)\in C\times\kappa$ and using the surjectivity of $\chi|_{gK}:gK\to C$, find a point $y\in gK$ with $\chi(y)=c$. Since the restriction $\bar\chi|_{\pi^{-1}(y)}:\pi^{-1}(y)\to\{c\}\times\kappa$ is bijective, there is a point $x\in \pi^{-1}(y)\subset \pi^{-1}(gK)=g\bar K$ with $\bar\chi(x)=(c,\alpha)$, so $\bar\chi|_{g\bar K}$ is surjective.

To see that it is injective, take any two distinct points $x,x'\in g\bar K$. If $\pi(x)=\pi(x')$, then for the point $y=\pi(x)=\pi(x')\in g\pi(K)=\pi(gK)$ the injectivity of the restriction $\bar \chi|_{\pi^{-1}(y)}$ implies that $\bar\chi(x)\ne\bar\chi(x')$. If $\pi(x)\ne \pi(x')$, then the injectivity of $\chi|_{gK}$ guarantees that $\chi(\pi(x))\ne\chi(\pi(x'))$ and then $\bar\chi(x)\ne\bar\chi(x')$ as $\bar\chi(x)\in\{\chi(\pi(x))\}\times\kappa$ and $\bar\chi(x)\in\{\chi(\pi(x'))\}\times\kappa$.
\smallskip

2. Given any element $g\in G$, we need to check that the restriction $\tilde \chi|_{g\tilde K}:g\bar K\to C$ is bijective. To see that  $\tilde\chi|_{g\tilde K}$ is surjective, observe that
$$\tilde \chi(g\tilde K)=\chi\circ \pi(g\tilde K)=\chi(g\,\pi(\tilde K))=\chi(gK)=C$$by the surjectivity of $\chi|_{gK}:gK\to C$.

To see that $\tilde\chi|_{g\tilde K}$ is injective, take any two distinct points $x,x'\in\tilde K=s(K)$ and observe $\pi(x)\ne \pi(x')$. Since $\pi$ is equivariant, $\pi(gx)=g\pi(x)\ne g\pi(x')=\pi(gx')$. Since $\pi(gx),\pi(gx')\in gK$ and $\chi|_{gK}$ is injective, $\tilde \chi(x)=\chi(\pi(gx))\ne \chi(\pi(gx'))=\tilde\chi(\pi(gx'))$ are we are done.
\end{proof}


Iterating the constructions from Proposition~\ref{p2.1}, we get the
{\em splitting construction} of kaleidoscopical configurations.

\begin{proposition}\label{p2.2} Let $X_0\to X_{1}\to \cdots \to X_m$ be a sequence of $G$-spaces linked by homogeneous $G$-equivariant mappings $\pi_i:X_i\to X_{i+1}$, $i<m$. Let $K_i\subset X_i$, $i\le m$, be subsets such that for every $i<m$ either the restriction $\pi_i|_{K_i}:K_i\to K_{i+1}$ is bijective or else $K_i=\pi_i^{-1}(K_{i+1})$. If the set $K_m$ is kaleidoscopic in the $G$-space $X_m$, then for every $i\le m$ the set $K_i$ is kaleidoscopic in the $G$-space $X_i$.
\end{proposition}

\begin{proof} This proposition can be derived from Proposition~\ref{p2.1} by the reverse induction on $i\in\{m,m-1,\dots,0\}$.
\end{proof}

Proposition~\ref{p2.2} can be alternatively written in terms of invariant equivalence relations.

Given an equivalence relation $E\subset X\times X$ on a set $X$ let
$X/E=\{[x]_E:x\in X\}$ be the quotient space consisting of the
equivalence classes $[x]_E=\{y\in X:(x,y)\in E\}$, $x\in X$. Denote
by $q_E:X\to X/E$, $q_E:x\mapsto [x]_E$,  the quotient mapping. For
a subset $K\subset X$ let $K/E=\{[x]_E:x\in K\}\subset X/E$ and
$[K]_E=\bigcup\limits_{x\in K}[x]_E\subset X$.

Let $E$ be an equivalence relation on a set $X$. A subset $K\subset X$ is  defined to be
\begin{itemize}
\item {\em $E$-parallel} \ if $K\cap [x]_E=[x]_E$ for all $x\in K$;
\item {\em $E$-orthogonal} \ if $K\cap [x]_E=\{x\}$ for all $x\in K$.
\end{itemize}
Given two equivalence relations $E\subset F$ on $X$ we can generalize these two notions defining $K\subset X$ to be
\begin{itemize}
\item {\em $F/E$-parallel} \ if $[K]_E\cap [x]_F=[x]_F$ for all $x\in K$;
\item {\em $F/E$-orthogonal} \ if $[K]_E\cap [x]_F=[x]_E$ for all $x\in K$.
\end{itemize}
Observe that a set $K\subset X$ is $E$-parallel ($E$-orthogonal) if and only if it is $E/\Delta_X$-parallel ($E/\Delta_X$-orthogonal). Here $\Delta_X=\{(x,x):x\in X\}$ stands for the smallest equivalence relation on $X$.

% For two equivalence relations $E\subset F\subset X^2$ on $X$ let $q^E_F:X/E\to X/F$ be the unique mapping such that $q_F^E\circ q_E=q_F$. This mapping assigns to each equivalence class $[x]_E$ the equivalence class $[x]_F=\big[[x]_E]_F$. Here for a subset $K\subset X$ we put $[K]_F=\bigcup_{x\in K}[x]_F$.

 An equivalence relation $E$ on a $G$-space $X$ is called
{\em $G$-invariant\/} if for each $(x,y)\in E$ and any $g\in G$ we get $(gx,gy)\in E$. For a $G$-invariant equivalence relation $E$ on $X$ the quotient space $X/E$ is a $G$-space under the induced action $$G\times X/E\to X/E,\;\;(g,[x]_E)\mapsto [gx]_E$$of the group $G$. In this case the quotient projection $q:X\to X/E$ is equivariant. $G$-Invariant equivalence relations on $G$-spaces are also called {\em imprimitivities}.

\begin{proposition}\label{p2.3} Let $\Delta_X=E_0\subset E_{1}\subset\dots\subset E_m$ be a sequence of $G$-invariant equivalence relations on a transitive $G$-space $X$. A subset $K\subset X$ is kaleidoscopic provided
\begin{enumerate}
\item the projection $K/E_m$ is kaleidoscopic in the $G$-space $X/E_m$;
\item for every $i<m$ the set $K$ is $E_{i+1}/E_i$-parallel or $E_{i+1}/E_i$-orthogonal.
\end{enumerate}
\end{proposition}

\begin{proof} For every $i\le m$ consider the $G$-space $X_i=X/E_i$ and the subset $K_i=K/E_i$ in $X_i$. Since $E_0=\Delta_X$, the space $X_0$ coincides with $X$. Next, for every $i<m$, consider the equivariant mapping $\pi_i:X_i\to X_{i+1}$, $\pi_i:[x]_{E_i}\mapsto [x]_{E_{i+1}}$. This mapping is homogeneous because of the transitivity of the $G$-space $X_i$.

We claim that the mappings $\pi_i$ satisfy the requirements of
Proposition~\ref{p2.2}. Indeed, if $K$ is $E_{i+1}/E_i$-parallel,
then  $K_i=\pi^{-1}_i(K_{i+1})$. If $K$ is $E_{i+1}/E_i$-orthogonal,
then the restriction $\pi_i|_{K_i}:K_i\to K_{i+1}$ is bijective.

Now Proposition~\ref{p2.2} implies that the set $K=K_0$ is kaleidoscopic in $X=X_0$.
\end{proof}


Proposition~\ref{p2.3} suggests the following notion that will be cenral in our subsequent discussion.

\begin{definition}\label{d2.4} A (kaleidoscopic) subset $K$ in a $G$-space $X$ is called {\em splittable} if there is an increasing sequence of  $G$-invariant equivalence relations $$\Delta_X=E_0\subset E_{1}\subset\dots\subset E_m=X\times X$$ such that for every $i<m$ the set $K$ is either $E_{i+1}/E_i$-parallel or $E_{i+1}/E_i$-orthogonal.
\end{definition}

Proposition~\ref{p2.3} implies that each splittable subset in a transitive $G$-space is kaleidoscopic. What about the inverse implication?

\begin{problem} For which $G$-spaces $X$, every kaleidoscopical configuration  $K\subset X$ is splittable?
\end{problem}



\section{Kaleidoscopical configurations in metric spaces}\label{s:prim}
Here we consider each metric space $(X,d)$ as a $G$-space endowed
with the natural action of its isometry group $G=\Iso(X)$. If this
action is transitive, then the metric space $X$ is called {\em
isometrically homogeneous}.

Let us recall that a metric space $(X,d)$ is {\em ultrametric} if the metric $d$ satisfies the strong triangle inequality
$$d(x,z)\le\max\{d(x,y),d(y,z)\}$$
for all $x,y,z\in X$. It follows that for every $\e\ge 0$ the relation
$$E_\e=\{(x,y)\in X^2:d(x,y)\le\e\}\subset X\times X$$ is an invariant equivalence relation on $X$.


\begin{theorem}\label{t2.7}
Let $(X,d)$ be an isometrically homogeneous ultrametric space with the finite distance scale $d(X\times X)=\{\varepsilon_0,\varepsilon_1,\ldots,\varepsilon_n\}$ where $0=\varepsilon_0<\varepsilon_1<\ldots<\varepsilon_n$. Then every kaleidoscopical configuration $K$ in $X$ is $(E_{\varepsilon_0},E_{\varepsilon_1},\ldots,E_{\varepsilon_n})$-splittable.
\end{theorem}

\begin{proof} Assume conversely that $K$ is not $(E_{\varepsilon_0},E_{\varepsilon_1},\ldots,E_{\varepsilon_n})$-splittable.
Then for some $k<n$ the set $K$ is neither $E_{\e_{k+1}}/E_{\e_k}$-parallel nor $E_{\e_{k+1}}/E_{\e_k}$-orthogonal. We can assume that $k$ is the smallest number with that property. By $[x]_{\e_i}$ we shall denote the closed $\e_i$-ball $[x]_{E_{\e_i}}$ centered at a point $x\in X$.

Since $K$ is not $E_{\e_{k+1}}/E_{\e_k}$-orthogonal, there are two points $u,v\in K$ such that $\e_k<d(u,v)=\e_{k+1}$.
Since $K$ is not $E_{\e_{k+1}}/E_{\e_k}$-parallel, there are points $w\in K$ and $z\in X$ such that $\e_{k}<\inf_{x\in K}d(z,x)=d(z,w)=\e_{k+1}$.

Since $X$ is isometrically homogeneous, we can find an isometry
 $\varphi:X\to X$ such that $\varphi(w)=z$. Then $\varphi([w]_{\e_{k}})=[z]_{\e_{k}}$ and we can define an isometry $\phi:X\to X$ letting
$$\phi(x)=\begin{cases}
\varphi(x) &  \mbox{ if $x\in[w]_{\e_{k}}$},\\
\varphi^{-1}(x) & \mbox{ if $x\in[z]_{\e_{k}}$},\\
x & \mbox{otherwise}.
\end{cases}$$
The isometry $\phi$ swaps the balls $[w]_{\e_k}$ and $[z]_{\e_k}$ but does not move points outside the union $[w]_{\e_k}\cup [z]_{\e_k}$. Since $K$ is $\chi$-kaleidoscopic, the restrictions $\chi|_{\phi(K)}$ and $\chi|_K$ are bijections onto $C$. Consequently, $\chi(w)=\chi(z')$ for some point  $z'\in[z]_{\e_k}$.
Taking into account that $d(w,z')=d(w,z)=\e_{k+1}=d(u,v)$ and $X$ is an isometrically homogeneous ultrametric space, we can construct an isometry $\psi:X\to X$ such that $\psi(u)=w$ and $\psi(v)=z'$. For this isometry, $w,z'\in\psi(K)$ and hence $\chi|_{\psi(K)}$ is not injective, contradicting the choice of the coloring $\chi$.
\end{proof}

\begin{problem}
Let $\{0,1\}^\omega$ be the Cantor space endowed with the standard ultrametric generating the product topology. Describe all kaleidoscopical configurations in $\{0,1\}^\omega$.
\end{problem}

\begin{remark} All closed kaleidoscopical configurations in $\{0,1\}^\omega$
can be characterized with usage of Theorem \ref{t2.7}. Among them
there are plenty of non-splittable configurations.
\end{remark}

A $G$-space $X$ is called {\em primitive} if each $G$-invariant
equivalence relation on $X$ is either $\Delta_X$ or $X\times X$.
Thus, each splittable configuration $K$ in a primitive $G$-space $X$
is trivial, i.e. either $K=X$ or $K$ is singleton. It is natural to
ask whether every kaleidoscopical configuration in a primitive
$G$-space is trivial?

The answer to this question is affirmative if $X$ is {\em 2-transitive} in the sense that for any pairs $(x,y),(x',y')\in X^2\setminus \Delta_X$ there is $g\in X$ such that $(x',y')=(gx,gy)$.

An example of a primitive $G$-space, which is not 2-transitive is
the  Euclidean space $\mathbb{R}^n$ of dimension $n\ge 2$ endowed
with the action  of its isometry group $\mathrm{Iso}(\mathbb{R}^n)$.
We show that $\IR^n$ contain $2^{\mathfrak c}$ unsplittable
kaleidoscopical configurations of cardinality $\mathfrak c$.

To construct a kaleidoscopic subset in $\IR^n$, use
Proposition~\ref{p1.10} and the following auxiliary definition.

Let $(X,d)$ be a metric space. By $S(x,r)=\{y\in X:d(x,y)=r\}$ we
denote the sphere of radius $r$ centered at a point $x\in X$.

\begin{definition} A subset $K$ of a metric space $(X,d)$ is called {\em rigid} if for any distinct points $x,y,z\in K$ and numbers $r_x,r_y,r_z\in d(K\times K)$ the spheres $S(x,r_x)$, $S(y,r_y)$, $S(z,r_z)$ have no common point in $X\setminus K$.
\end{definition}

\begin{theorem}\label{t:indep} Let $X$ be a metric space and $G\subset\Iso(X)$ be a group of isometries of $X$. Each infinite rigit subset $K\subset X$ of cardinality $|K|\ge|G|$ is kaleidoscopical.
\end{theorem}

\begin{proof}  The kaleidoscopicity of the set $K$ will follow from Proposition~\ref{p1.10} as soon as we check that the  hypergraph $(V,\F)=(X,\{gK:g\in G\})$ satisfies the conditions (1)--(2) for the cardinal $\kappa=|K|$. Since $|G|\le \kappa=|K|=|gK|$ for all $g\in G$, the condition (1) is satisfied.

To show that (2) holds, take any subset $A\subset G$ of cardinality
$|A|<\kappa$ and any subset $B\in X\setminus AK$ of cardinality
$|B|<\kappa$. We need to show that $|\St(B,\F)\cap AK|<\kappa$. This
will follow from $\max\{|A|,|B|\}<\kappa$ as soon as we check that
$|\St(b,\F)\cap aK|\le 2$ for every $b\in B$ and $a\in A$. Assuming
conversely that $\St(b,\F)\cap aK$ contains three pairwise distinct
points $x,y,z$ we shall obtain a contradiction with rigidity of $K$
because $d(b,x),d(b,y),d(b,z)\in d(K\times K)$ and $b$ is the common
point of the spheres $S(x,d(b,x))$, $S(y,d(b,y))$, $S(z,d(b,z))$.
\end{proof}

To apply Theorem~\ref{t:indep}, we need an effective construction of
rigid subsets in metric spaces.  

\begin{lemma}\label{l3.6} Any algebraically  independent over $\IQ$ subset $A$ of an affine line (identified with $\IR$) in the Euclidean space $\IR^n$ is rigid.
\end{lemma}

\begin{proof} Let $a,b,c$ be pairwise distinct points of $A$, $b\in[a,c]$, $x\in\IR^n\setminus A$ and $d(x,a)=r_a$, $d(x,b)=r_b$, $d(x,c)=r_c$. Since $\cos(\angle abx)=-\cos(\angle cbx)$, by the cosines theorem, we get
\begin{itemize}
\item[$(*)$] $(c-b)(b-a)^2+(c-b)r_b^2-(c-b)r_a^2+(b-a)(c-b)^2+(b-a)r_b^2-(b-a)r_c^2=0$.
\end{itemize}
Assuming $r_a,r_b,r_c\in d(A,A)$ and taking into account that at most two of the three numbers $r_a,r_b,r_c$ can be equal, after corresponding substitutions and opening all brackets in $(\ast)$, we get a contradiction with algebraic independence of $A$.
\end{proof}

Now we are able to prove the promised:

\begin{theorem} For $n>1$ the Euclidean space $\IR^n$ contains $2^{\mathfrak c}$ many kaleidoscopic subsets.
\end{theorem}
\begin{proof} Apply Theorem~\ref{t:indep} and Lemma~\ref{l3.6}.
\end{proof}

\begin{problem} Does the Euclidean space $\IR^n$ of dimension $n\ge 2$ contain a non-trivial finite or countable kaleidoscopical subset?
\end{problem}
If such a set $K$ exists, then its cardinality $|K|$ is not less
that the chromatic number of $\IR^n$.

We recall that the {\em chromatic number} $\chi(X)$ of a metric
space $X$ is equal to the smallest number $\kappa$ of colors for
which there is a coloring of $X$ without monochrome points at the
distance $1$. It is known that $4\le \chi(\IR^2)\le 7$ but the exact
value of $\chi(\IR^2)$ is not known. There is a conjecture that
$\chi(\IR^n)=2^{n+1}-1$, see \cite[\S47]{Soifer}.

\begin{problem} Is every finite kaleidoscopical configuration in a
(finite) primitive $G$-space trivial?
\end{problem}
Some examples of infinite $G$-spaces with only trivial finite
kaleidoscopical configurations can be found in \cite[Chapter 8]{BP}

A space $\IR^n$ can also be considered as a $G$-space with respect
to the group $G=\Aff(\IR^n)=\{\lambda
x+a:\lambda\in\IR\setminus\{0\},a\in\IR^n\}$ of all affine
transformations. The kaleidoscopical configurations $K$ of
cardinality $|K|<\mathfrak{c}$ in this space are singletons because
any line that contains more then one point of a kaleidoscopical
configuration has no distinct points of the same color. On the other
hand, every affine subspace of $\IR^n$ is kaleidoscopical. Moreover,
using Proposition~\ref{p1.10}, we can construct $2^\mathfrak{c}$
non-splittable affine kaleidoscopical configuration of size
$\mathfrak{c}$ in $\IR^n$ for $n>1$.

Restricting ourself with only translations of $\IR^n$, we get a
kaleidoscopical configuration of any size $\kappa$,
$1\le\kappa\le\mathfrak{c}$. It follows from well-known
decomposition of $\IR^n$ in the direct sum of rationals and the
observation that $\IZ$ has a kaleidoscopical configuration of any
finite size.
\section{Haj\'os properties in groups and $G$-spaces}\label{s:Hajos}

In this section we reveal the relation of splittability of
kaleidoscopical configurations in finite Abelian groups to the
Haj\'os property introduced in \cite{Hajos} and studied in
\cite{Sands62}, \cite{Sz}, \cite{SS}.

We recall that an Abelian group $G$ has the {\em Haj\'os property} if for each factorization $G=AB$ either $A$ or $B$ is periodic. A subset $A$ of a group $G$ is called {\em periodic} if $A=gA$ for some non-identity element $g\in G$.  Finite Abelian groups with Haj\'os property were classified in \cite{Sands62}:

\begin{theorem}[Haj\'os-Sands]\label{HS} A finite Abelian group $G$ has the Haj\'os property if and only if $G$ is isomorphic to a subgroup of a group that has one of the following types:

$(p^n,q)$, $(p^2,q^2)$, $(p^2,q,r)$, $(p,q,r,s)$, $(p,p)$, $(p,3,3)$, $(3^2,3)$,

$(p^3,2,2)$, $(p^2,2,2,2)$, $(p,2^2,2)$, $(p,2,2,2,2)$, $(p,q,2,2)$, $(2^n,2)$, $(2^2,2^2)$,

\noindent where $p<q<r<s$ are distinct primes and $n\in\IN$.
\end{theorem}

A group $G$ is of type $(n_1,\dots,n_k)$ if $G$ is isomorphic to the direct sum of cyclic groups $C_{n_1}{\oplus}\cdots{\oplus}\; C_{n_k}$.

Now let us define two weakenings of the Haj\'os property.

\begin{definition}\label{d3.2} An Abelian group $G$ is defined to have
\begin{itemize}
\item the {\em semi-Haj\'os property} if each complemented subset $A\subsetneq G$ either is periodic or has a periodic complementer factor in $G$;
\item the {\em demi-Haj\'os property} if for each factorization $G=A B$ one of the factors $A,B$ either is periodic or has a periodic complementer factor.
\end{itemize}
\end{definition}

It is clear that for each Abelian group $G$
$$\mbox{Haj\'os } \Ra \mbox{ semi-Haj\'os }\Ra\mbox{ demi-Haj\'os}.$$

\begin{problem} Is the semi-Haj\'os property of finite Abelian groups equivalent to the demi-Haj\'os property?
\end{problem}

The demi-Haj\'os property was (implicitly) defined in  \cite{Sands74} and follows from the quasi-periodicity of any factorization of the group.
In contrast to the Haj\'os property, at the moment we have no classification of finite Abelian groups possessing the demi-Haj\'os property. It is even not known if each finite cyclic group has the demi-Haj\'os property, see Problem 5.4 in \cite{SS}. The best known positive result on the semi-Haj\'os property is the following version of Theorem 5.13 \cite{SS}:

\begin{theorem}[Szab\'o, \cite{Sz2}]\label{semiH} Each finite Abelian group $G$ of square-free order $|G|$ has the semi-Haj\'os property.
\end{theorem}

We say that a number $n$ is {\em square-free} if $n$ is not divisible by the square $p^2$ of any prime number $p$.

Surprisingly, the following problem of Fuchs and Sands
\cite[p.364]{Fuchs}, \cite{Sands74}, \cite[p.120]{SS} posed in 60-ies still is open:

\begin{problem} Has each finite Abelian group the demi-Haj\'os property?
\end{problem}

The ``semi'' version of this problem also is open:

\begin{problem}\label{pr3.4} Has each finite Abelian group the semi-Haj\'os property?
\end{problem}

The semi-Haj\'os property is tightly connected with the splittability of kaleidoscopical configurations. In order to state the precise result, let us generalize the definition of the semi-Haj\'os property to $G$-spaces.

\begin{definition} A $G$-space $X$ has the {\em semi-Haj\'os property} if  for each kaleidoscopical subset $K\subsetneq X$ there is a $G$-invariant equivalence relation $E\ne\Delta_X$ on $X$ such that $K$ is $E$-parallel or $E$-orthogonal and the set $K/E$ is kaleidoscopical in the $G$-space $X/E$.
\end{definition}

For finite Abelian groups this definition of the semi-Haj\'os property agrees with that given in Definition~\ref{d3.2}.

\begin{proposition} A finite Abelian group $G$ has the semi-Haj\'os property if and only if it has that property as a $G$-space.
\end{proposition}

\begin{proof} Assume that the group $G$ has the semi-Haj\'os property.
To show that the $G$-space $G$ has the semi-Haj\'os property, take
any kaleidoscopical subset $A\subset G$. By Corollary~\ref{c1.5},
$A$ is complementable and hence has a complementer factor $B$. Since
$G$ has the semi-Haj\'os property, either $A$ is periodic or else
$A$ has a periodic complementer factor. In the latter case we can
assume that the complementer factor $B$ is periodic. Consequently
there is a non-trivial cyclic subgroup $H\subset G$ such that either
$A+H=A$ or $B+H=B$. Consider the quotient group $G/H$ and the
quotient homomorphism $q:G\to G/H$. By Lemma 2.6 of~\cite{SS}, the
images $A/H=q(A)$ and $B/H=q(B)$ form a factorization $G/H=A/H\cdot
B/H$ of the quotient group $G/H$. Consequently, the set $A/H$ is
complemented in $G/H$ and by Corollary~\ref{c1.5}, it is
kaleidoscopical in $G/H$.

The subgroup $H$ induces a $G$-invariant equivalence relation $E=\{(x,y)\in G:x-y\in H\}$ whose quotient space $G/E$ coincides with the quotient group $G/H$. We claim that the set $A$ is either $E$-parallel or $E$-orthogonal. By the choice of the group $H$, we get $A=A+H$ or $B=B+H$. In the first case the set $A$ is $E$-parallel. In the second case $A$ is $E$-orthogonal as $(A-A)\cap H\subset (A-A)\cap (B-B)=\{0\}$.
\smallskip

Now assuming that the $G$-space $G$ has the semi-Haj\'os property,
we shall prove that the group $G$ has the semi-Haj\'os property.
Given any complemented subset $A\subset G$ we need to show that
either $A$ is periodic or else $A$ has a periodic complementer
factor. By Corollary~\ref{c1.5}, the set $A$ is kaleidoscopical in
the $G$-space $G$. The semi-Haj\'os property of the $G$-space $G$
guarantees the existence of a $G$-invariant equivalence relation
$E\ne\Delta_G$ on $G$ such that $A$ is $E$-parallel or
$E$-orthogonal and $A/E$ is kaleidoscopical in $G/E$. It follows
that the equivalence class $H=[0]_E$ of zero is a subgroup of the
group $G$. Taking into account that $E$ is $G$-invariant, we
conclude that $(x,y)\in E$ iff $x-y\in[0]_E$. So, $G/E$ coincides
with the quotient group $G/H$. The set $A/H$, being kaleidoscopical,
is complemented in $G/H$ according to Corollary~\ref{c1.5}.
Consequently, there is a subset $B_H\subset G/H$ such that
$G/H=A/H\cdot B_H$. Let $q:G\to G/H$ be the quotient mapping and
$s:G/H\to G$ be any section of $q$.

Now consider two cases. If $A$ is $E$-parallel, then $A=A+H$ is periodic and complemented as $B=s(B_H)$ is a complementer factor to $A$ in $G$. If $A$ is $E$-orthogonal, then the complete preimage $B=q^{-1}(B_H)$ is a periodic complementer factor to $A$ in $G$.
\end{proof}


Now we reveal the relation between the semi-Haj\'os property and the
splittability of kaleidoscopical sets.

\begin{proposition}\label{p3.7} If each kaleidoscopical subset of a transitive $G$-space $X$ is splittable, then $X$ has the semi-Haj\'os property.
\end{proposition}

\begin{proof} To show that $X$ has the semi-Haj\'os property, fix any kaleidoscopical subset $K\subset X$. By our assumption, $K$ is $(E_0,\dots,E_m)$-splittable by some increasing chain of invariant equivalence relations $\Delta_X=E_0\subset \cdots\subset E_m=X\times X$.  For every $i\le m$ consider the quotient $G$-space $X_i=X/E_i$ and let $q_i:X\to X_i$ be the quotient projection. Also let $K_i=q_i(K)\subset X_i$. By Proposition~\ref{p2.2}, $K_i$ is kaleidoscopical in the $G$-space $X_i$.
In particular, $K_1$ is kaleidoscopical in $X_1=X/E_1$. By
Definition~\ref{d2.4}, $K=K_0$ is either $E_1$-parallel or
$E_1$-orthogonal. This means that $X$ has the semi-Haj\'os property.
\end{proof}

Theorems~\ref{t2.7} and Proposition~\ref{p3.7} imply

\begin{corollary} Each isometrically homogeneous ultrametric space with finite distance scale has the semi-Haj\'os property.
\end{corollary}

A $G$-space $Y$ is defined to be a {\em quotient} of a $G$-space $X$
if  $Y$ is the image of $X$ under a $G$-equivariant mapping $f:X\to
Y$.

\begin{proposition}\label{p3.9} Each kaleidoscopical subset of a $G$-space $X$ is splittable provided that:
\begin{enumerate}
\item each quotient $G$-space of $X$ has the semi-Haj\'os property and
\item $X$ admits no strictly increasing infinite sequence $(E_n)_{n\in\w}$ of $G$-invariant equivalence relations.
\end{enumerate}
\end{proposition}

\begin{proof} Assume that some kaleidoscopical subset $K\subset X$ is not splittable. Let $K_0=K$, $E_0=\Delta_X$, and $X_0=X/E_0=X$. Since $X$ has the semi-Haj\'os property, there is a $G$-invariant equivalence relation
$E_1\ne \Delta_X$ on $X_0$ such that the set $K_1=K_0/E_0$ is
kaleidoscopical in the $G$-space $X_1=X_0/E_1$ and $K_0$ is either
$E_1$-parallel or $E_1$-orthogonal.

By our assumption, $K$ is not splittable, so $X_1$ is not a
singleton. The $G$-space $X_1=X/E_1$, being a quotient of $X$, has
the semi-Haj\'os property. Consequently, for the kaleidoscopical set
$K_1\subset X_1$ there is a $G$-invariant equivalence relation
$\tilde E_2\ne\Delta_{X_1}$ on $X_1$ such that the set $K_1$ is
$\tilde E_2$-parallel or $\tilde E_2$-orthogonal and the quotient
set $K_2=K_1/\tilde E_1$ is kaleidoscopical in the $G$-space
$X_2=X_1/\tilde E_1$. Let $q^1_2:X_1\to X_2$ be the quotient
projection. The composition $q^1_2\circ q_1:X\to X_2$ determines the
$G$-invariant equivalence relation $E_2=\{(x,x')\in X^2:q^1_2\circ
q_1(x)=q^1_2\circ q_1(x')\}$ on $X$ such that $X/E_2=X_2$ and
$K_2=K/E_2$ and $K_1$ is either $E_2/E_1$-parallel or
$E_2/E_1$-orthogonal.

Continuing by induction, we shall produce an infinite increasing
sequence $(E_n)_{n\in\w}$ of $G$-invariant equivalence relations on
$X$ such that for every $n\in\IN$ the set $K_n=K/E_n$ is
kaleidoscopical in the $G$-space $X/E_n$ and $K$ is either
$E_{n}/E_{n-1}$-parallel or $E_n/E_{n-1}$-orthogonal. But the
existence of an infinite strictly increasing sequence of
$G$-invariant equivalence relations on $X$ contradicts our
assumption.
\end{proof}

Since each quotient group of a finite Abelian group $G$
is isomorphic to a subgroup of $G$, Proposition~\ref{p3.9} implies:

\begin{corollary}\label{c3.10} If each subgroup of a finite Abelian group $G$ has the semi-Haj\'os property, then each kaleidoscopical subset $K\subset G$ is  splittable.
\end{corollary}

\begin{question}\label{q4.13} Assume that a finite Abelian group $G$ has  the semi-Haj\'os property. Has each subgroup of $G$ that property?
\end{question}

The classification of finite Abelian groups with Haj\'os property
given in Theorem~\ref{HS} implies that this property is inherited by subgroups. Because of that, Corollary~\ref{c3.10} implies:

\begin{corollary} For a finite Abelian group $G$ with the Haj\'os property, each kaleidoscopical subset $K\subset G$ is splittable.
\end{corollary}

Also Proposition~\ref{p3.9} and Theorem~\ref{semiH} imply:

\begin{corollary} For a finite Abelian group $G$ of square-free order $|G|$ each kaleidoscopical subset $K\subset G$ is splittable.
\end{corollary}


\begin{remark} It follows from Proposition~\ref{p3.7} and Corollary~\ref{c3.10} that Question~\ref{q4.13} and Problem~\ref{pr3.4} are equivalent (and both are open and apparently difficult).
\end{remark}

According to an old result of Haj\'os \cite{Hajos}, if in a factorization  $\IZ=A+B$ of the infinite cyclic group $\IZ$ the factor $A$ is finite, then the factor $B$ is periodic. We do not know if the same is true for the groups $\IZ^n$ with $n\ge 2$.

{\bf Acknowledgement.} We thank referee for remarks, suggestions, and references.

\begin{thebibliography}{99}

\bibitem{PB} M.~Aigner, G.~Ziegler, {\em Proofs from The Book}, Springer-Verlag, Berlin, 2010.

\bibitem{Hajos} G.~Haj\'os, {\em Sur la factorisation des groupes ab\`eliens}, \v Casopis P\v est. Mat. Fys. {\bf 74} (1949), 157--162.

\bibitem{Fuchs}  J.M.~Irwin, E.A.~Walker, {\em Topics in Abelian Groups}, Glenview, Illinois, 1963.

\bibitem{BP} I.~Protasov, T.~Banakh, {\it Ball Structures and
Colorings of Graphs and Groups}, Math. Stud. Monogr. Ser, Vol. 11, VNTL Publisher, Lviv, 2003.

\bibitem{PP} I.~Protasov, K.~Protasova, {\it Kaleidoscopical Graphs and Hamming codes}, Voronoi's Impact on Modern Science, Book 4, Volume 1, Proc. 4th Intern. Conf. on Analytic Number Theory and Spatial Tesselations, Institute of Mathematics, NAS of Ukraine, Kyiv, 2008, 240--245.

\bibitem{PO} K.D.~Protasova, \emph{Kaleidoscopical Graphs},
Math. Stud., \textbf{18} (2002), 3--9.

\bibitem{Ryser} H.J.~Ryser, {\em A combinatorial theorem with an application to latin rectangles}, Proc. Amer. Math. Soc. {\bf 2} (1951), 550--552.

\bibitem{Sands62} A.D.~Sands, {\em On the factorisation of finite Abelian groups II}, Acta Math. Acad. Sci. Hungar. {\bf 13} (1962), 153--169.

\bibitem{Sands74} A.D.~Sands, {\em On a conjecture of G.~Haj\'os}, Glasgow Math. J. {\bf 15} (1974), 88--89.

\bibitem{Soifer} A.~Soifer, {\em The mathematical coloring book}, Springer, 2009.

\bibitem{Sz2} S.~Szab\'{o}, {\em Groups admitting only quasi-periodic factorizations}, Annales Sci. Budapest., Sect. Comp. 29 (2008), 239--243.

\bibitem{Sz} S.~Szab\'{o}, {\em Topics in Factorization of Abelian Groups}, Birkh\"auser Basel, 2005.

\bibitem{SS} S.~Szab\'{o}, A.~Sands, {\em Factoring groups into subsets}, CRC Press, 2009.
\end{thebibliography}

\end{document}