The chromatic thresholds of graphs

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].