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 ${dist}(a,b)\leq d$ for all $a\in A$ and $b\in B$, or ${dist}(a,b)>d$ 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.

Keywords

  1. graphs
  2. sparsity
  3. Erdös--Hajnal conjecture
  4. weak coloring numbers
  5. graph power

MSC codes

  1. 05C99

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 447 - 464
ISSN (online): 1095-7146

History

Submitted: 4 June 2020
Accepted: 24 November 2020
Published online: 23 March 2021

Keywords

  1. graphs
  2. sparsity
  3. Erdös--Hajnal conjecture
  4. weak coloring numbers
  5. graph power

MSC codes

  1. 05C99

Authors

Affiliations

Funding Information

Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 677651

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