The chromatic thresholds of graphs

TL;DR

This paper determines the exact chromatic thresholds for all graphs, solving longstanding questions and confirming conjectures about the relationship between graph structure and chromatic number.

Contribution

It characterizes the chromatic thresholds for all graphs and proves the exact values, answering a question by Erdős and Simonovits and confirming a conjecture by Łuczak and Thomassé.

Findings

Chromatic thresholds are in a specific set of rational numbers.

Complete characterization of graphs by their chromatic thresholds.

Resolved a 50-year-old open problem in graph theory.

Abstract

The chromatic threshold delta_chi(H) of a graph H is the infimum of d>0 such that there exists C=C(H,d) for which every H-free graph G with minimum degree at least d|G| satisfies chi(G)<C. We prove that delta_chi(H) \in {(r-3)/(r-2), (2r-5)/(2r-3), (r-2)/(r-1)} for every graph H with chi(H)=r>2. We moreover characterise the graphs H with a given chromatic threshold, and thus determine delta_chi(H) for every graph H. This answers a question of Erd\H{o}s and Simonovits [Discrete Math. 5 (1973), 323-334], and confirms a conjecture of {\L}uczak and Thomass\'e [preprint (2010), 18pp].