Perfect cliques and $G_\delta$ colorings of Polish spaces

Abstract:

A coloring of a set $X$ is any subset $C$ of $[X]^N$, where $N>1$ is a natural number. We give some sufficient conditions for the existence of a perfect $C$-homogeneous set, in the case where $C$ is $G_\delta$ and $X$ is a Polish space. In particular, we show that it is sufficient that there exist $C$-homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect $3$-clique if it contains any uncountable $k$-clique, where $k$ is a natural number or $\aleph _0$ (a set $K$ is a $k$-clique in $X$ if the convex hull of any of its $k$-element subsets is not contained in $X$).