Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes

TL;DR

This paper proves that in triangle-free graphs with large chromatic number, there exist induced cycles (holes) of any specified number of consecutive lengths, revealing a structural property related to graph coloring.

Contribution

It establishes that large chromatic number in triangle-free graphs guarantees the presence of consecutive-length holes, extending understanding of graph structure.

Findings

Large chromatic number implies existence of consecutive holes

Holes of any specified length can be found in such graphs

Supports conjectures about structure of high chromatic number graphs

Abstract

A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.