Erd\H{o}s-Hajnal properties for powers of sparse graphs

TL;DR

This paper establishes Erdős-Hajnal type properties for powers of graphs within nowhere dense classes, demonstrating large homogeneous pairs with controlled distances, and extends these results to First-Order interpretations.

Contribution

It proves Erdős-Hajnal properties for powers of sparse graphs in nowhere dense classes and generalizes to First-Order interpreted graph classes.

Findings

Existence of large homogeneous pairs with bounded or unbounded distances

Extension of properties to First-Order interpretations

Results hold for any nowhere dense class and fixed distance d

Abstract

We prove that for every nowhere dense class of graphs $\mathcal{C}$, positive integer $d$, and $\varepsilon>0$, the following holds: in every $n$-vertex graph $G$ from $\mathcal{C}$ one can find two disjoint vertex subsets $A,B\subseteq V(G)$ such that $|A|\geq (1/2-\varepsilon)\cdot n$ and $|B|=\Omega(n^{1-\varepsilon})$ and either $\operatorname{dist}(a,b)\leq d$ for all $a\in A$ and $b\in B$, or $\operatorname{dist}(a,b)>d$ for all $a\in A$ and $b\in B$. We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dense graph classes.