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The chromatic threshold δ χ (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 thatfor every graph H with χ(H) = r 3. We moreover characterise the graphs H with a given chromatic threshold, and thus determine δ χ (H) for every graph H. This answers a question of Erdős and Simonovits [Discrete Math. 5 (1973), 323-334], and confirms a conjecture of Luczak and Thomassé [preprint (2010), 18pp]. δ χ (H) := inf d : ∃ C = C(H, d) such that if G is a graph on n vertices, with δ(G) dn and H ⊆ G, then χ(G) C .

We consider the binomial random graph G n,p and determine a sharp threshold function for the edge-Ramsey propertyfor all 1 , . . . , r , where C denotes the cycle of length . As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above.