Induced subgraphs of graphs with large chromatic number. III. Long holes

TL;DR

This paper proves a 1985 conjecture by Gyárfás, showing that graphs with large chromatic number necessarily contain either a large clique or a long induced cycle, advancing understanding of graph structure.

Contribution

It confirms Gyárfás's conjecture for all parameters, establishing a fundamental link between chromatic number and induced cycle length.

Findings

Graphs with large chromatic number contain long induced cycles.

Large chromatic number implies the existence of large cliques or long holes.

The conjecture holds for all values of k and ℓ.

Abstract

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$.