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.
Highlights
The theory of sparse graphs concentrates on defining and investigating combinatorial measures of sparsity for graphs, as well as more complicated relational structures
We shall say that C is strongly uniformly limit quasi-wide (SULQW) with margin s if for every d ∈ N, every graph G ∈ C, and every infinite set W ⊆ V (G), there exists a set S ⊆ V (G) with |S| s(d) such that in G − S one can find an infinite d-scattered set the electronic journal of combinatorics 23(2) (2016), #P2.32 contained in W
We remark that if one is only interested in the implication from limit nowhere denseness to uniform limit quasi-wideness, there is no need of using the slightly stronger assumption provided by Lemma 3.1 — the assumption Kω ∈/ C ω suffices
Summary
The theory of sparse graphs concentrates on defining and investigating combinatorial measures of sparsity for graphs, as well as more complicated relational structures. We investigate three definitions of nowhere denseness that are known to be equivalent: (1) the classic definition, both using shallow minors and shallow topological minors, (2) quasi-wideness, (3) the definition via the splitter game due to Grohe et al [3] We show that these notions have natural limit variants that are conceptually simpler, and (with some technical caveats) can be proved to be equivalent to the standard ones using Los’s theorem. Limit quasi- wideness can be defined as follows: for every natural number d and every infinite graph G ∈ C , one can delete a finite number of vertices from G so that the remaining graph contains an infinite set of vertices that are pairwise at distance at least d from each other Note that this the electronic journal of combinatorics 23(2) (2016), #P2.32 definition is fully formal, and vague intuitions of small/large from the discrete case are replaced by finite/infinite.
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