On ultralimits of sparse graph classes

TL;DR

This paper explores the properties of nowhere dense graph classes through ultraproduct limit objects, revealing simpler conceptual frameworks that unify various definitions in the theory of sparse graphs.

Contribution

It introduces a novel approach to analyze nowhere dense classes using ultraproducts, simplifying the understanding of their properties and equivalences.

Findings

Equivalent definitions of nowhere denseness correspond to natural properties of limit objects

Ultraproduct-based limits provide simpler reasoning tools

The approach unifies different characterizations of sparse graph classes

Abstract

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. It appears that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, correspond to natural notions for the limit objects that are conceptually simpler and allow for less technically involved reasonings.