Constraint satisfaction problems, compactness and non-measurable sets

TL;DR

This paper explores the concept of compactness in finite relational structures, linking it to set theory and the existence of non-measurable sets, with implications for understanding the foundations of mathematics.

Contribution

It establishes a connection between the compactness of relational structures and set-theoretic properties, showing that non-measurable sets arise in certain cases.

Findings

Compactness for width-one structures is provable in ZF set theory.

Non-measurable sets in 3-space are implied for structures with greater width.

The work bridges finite model theory and set theory, revealing foundational implications.

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.