The chromatic thresholds of graphs

The chromatic threshold δχ(H)δχ(H) of a graph HH is the infimum of d>0d>0 such that there exists C=C(H,d)C=C(H,d) for which every HH-free graph GG with minimum degree at least d|G|d|G| satisfies χ(G)⩽Cχ(G)⩽C. We prove that δχ(H)∈{r−3r−2,2r−52r−3,r−2r−1} for every graph HH with χ(H)=r⩾3χ(H)=r⩾3. We moreover characterise the graphs HH with a given chromatic threshold, and thus determine δχ(H)δχ(H) for every graph HH. This answers a question of Erdős and Simonovits [P. Erdős, M. Simonovits, On a valence problem in extremal graph theory, Discrete Math. 5 (1973), 323–334], and confirms a conjecture of Łuczak and Thomassé [Tomasz Łuczak, Stéphan Thomassé, Colouring dense graphs via VC-dimension, arXiv:1011.4310 (submitted for publication)].