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 Ne\v{s}et\v{r}il and R\"odl (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 function of $k$ and $r$. This is comparable to the known lower bound $2^{\Omega(rk)}$. Our proof relies on a recent result of Saxton and Thomason (2015) (or, alternatively, on a recent result of Balogh, Morris, and Samotij (2015)) from which we deduce a quantitative version of Ramsey's theorem in random graphs.