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arXiv · 2006.01500

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

Marcin Briański·Piotr Micek·Michał Pilipczuk·Michał T. Seweryn
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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.

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Marcin Briański, Piotr Micek, Michał Pilipczuk, Michał T. Seweryn. 2020-06-02. Erd\H{o}s-Hajnal properties for powers of sparse graphs. https://arxiv.org/abs/2006.01500

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