Non-Homogenizable Classes of Finite Structures

TL;DR

This paper establishes a necessary condition for the homogenization of classes of finite structures, demonstrating that certain classes, including some CSP instances and MSO-definable structures, are not homogenizable, thus clarifying the boundaries of homogenization.

Contribution

It provides the first necessary condition for homogenization of finite structure classes, identifying key non-homogenizable classes and clarifying the limits of existing homogenization techniques.

Findings

Certain classes of finite structures are not homogenizable.

Homogenization distinguishes CSPs solvable by local consistency.

Treewidth two is a dividing line for homogenizability in MSO-definable classes.

Abstract

Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability.