Abstract
The chromatic threshold δχ(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 χ(G)⩽C. We prove that δχ(H)∈{r−3r−2,2r−52r−3,r−2r−1} for every graph H with χ(H)=r⩾3. We moreover characterise the graphs H with a given chromatic threshold, and thus determine δχ(H) for every graph H. 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)].
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