# Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes

## TL;DR

This paper proves a conjecture that graphs with bounded clique number and large chromatic number necessarily contain long odd cycles, advancing understanding of graph structure related to chromatic properties.

## Contribution

It confirms Gyarfas's conjecture, establishing that such graphs always contain long odd holes, linking chromatic number to the existence of specific cycle lengths.

## Key findings

- Graphs with bounded clique number and large chromatic number have long odd holes.
- The proof confirms a longstanding conjecture in graph theory.
- Provides new insights into the structure of complex graphs.

## Abstract

We prove a conjecture of Andras Gyarfas, that for all k,t, every graph with clique number at most k and sufficiently large chromatic number has an odd hole of length at least t.