Constraint satisfaction problems, compactness and non-measurable sets

TL;DR

It is shown that if A has width one, then the compactness of A can be proved in the axiom system of Zermelo and Fraenkel, but otherwise, the compactness of A implies the existence of non-measurable sets in 3-space.

Abstract

A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to A. We show that if A has width one, then the compactness of A can be proved in the axiom system of Zermelo and Fraenkel, but otherwise, the compactness of A implies the existence of non-measurable sets in 3-space.

Theorems & Definitions (8)

• Theorem 1.1: ktv
• Theorem 1.2
• Lemma 4.1
• proof
• Lemma 4.2
• Lemma 4.3
• proof
• proof : Proof of Theorem \ref{['main']}