On Ultralimits of Sparse Graph Classes

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. Our goal is to demonstrate 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. In particular, using our approach we are able to give alternative proofs of several fundamental facts about nowhere denseness, which are less technically involved than the ones that can be found in the literature.