Linear-sized minors with given edge density

TL;DR

The paper proves that graphs with sufficiently high chromatic number contain minors with a linear number of vertices and high edge density, extending bounds related to Hadwiger's conjecture and graph minors.

Contribution

It establishes a bound linking chromatic number to minors with specified size and density, building on recent work and improving previous bounds.

Findings

Graphs with large chromatic number contain dense minors of linear size.

The bound depends logarithmically on the inverse of the density parameter.

Extends recent results on Hadwiger's conjecture and graph minors.

Abstract

It is proved that for every $\varepsilon>0$, there exists $K>0$ such that for every integer $t\ge2$, every graph with chromatic number at least $Kt$ contains a minor with $t$ vertices and edge density at least $1-\varepsilon$. Indeed, building on recent work of Delcourt and Postle on linear Hadwiger's conjecture, for $\varepsilon\in(0,\frac{1}{256})$ we can take $K=C\log\log(1/\varepsilon)$ where $C>0$ is a universal constant, which extends their recent $O(t\log\log t)$ bound on the chromatic number of graphs with no $K_t$ minor.