Reconstruction of a coloring from its homogeneous sets

TL;DR

This paper investigates the conditions under which a coloring of a set can be uniquely reconstructed from its homogeneous subsets, including a Borel method for such reconstruction, and explores both reconstructibility and non-reconstructibility scenarios.

Contribution

It introduces criteria for reconstructibility of colorings from homogeneous sets and demonstrates a Borel-based reconstruction method, advancing understanding in combinatorial and descriptive set theory.

Findings

Reconstructibility depends on specific conditions of the coloring.

A Borel method exists for reconstructing colorings from homogeneous sets.

The paper identifies cases where reconstruction is impossible.

Abstract

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