Reconstruction of a coloring from its homogeneous sets

Abstract

We study a reconstruction problem for colorings. Given a finite or countable set $X$, a coloring on $X$ is a function $φ: [X]^{2}\to \{0,1\}$, where $[X]^{2}$ is the collection of all 2-elements subsets of $X$. A set $H\subseteq X$ is homogeneous for $φ$ when $φ$ is constant on $[H]^2$. Let $hom(φ)$ be the collection of all homogeneous sets for $φ$. The coloring $1-φ$ is called the complement of $φ$. We say that $φ$ is {\em reconstructible} up to complementation from its homogeneous sets, if for any coloring $ψ$ on $X$ such that $hom(φ)=hom(ψ)$ we have that either $ψ=φ$ or $ψ=1-φ$. We present several conditions for reconstructibility and non reconstructibility. We show that there is a Borel way to reconstruct a coloring from its homogeneous sets.