An exponential-type upper bound for Folkman numbers

For given integers k and r, the Folkman number f(k;r) is the smallest number of vertices in a graph G which contains no clique on k+1 vertices, yet for every partition of its edges into r parts, some part contains a clique of order k. The existence (finiteness) of Folkman numbers was established by Folkman (1970) for r=2 and by Nešetřil and Rödl (1976) for arbitrary r, but these proofs led to very weak upper bounds on f(k;r).Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on f(k;2). Here, we establish a further improvement by showing an upper bound on f(k;r) which is exponential in a polynomial of k and r. This is comparable to the known lower bound 2Ω(rk). Our proof relies on a recent result of Saxton and Thomason (or, alternatively, on a recent result of Balogh, Morris, and Samotij) from which we deduce a quantitative version of Ramsey’s theorem in random graphs.