A lower bound for set-coloring Ramsey numbers

The set-coloring Ramsey number R-r,R-s(k) is defined to be the minimum n such that if each edge of the complete graph K-n is assigned a set of s colors from {1, ... ,r}, then one of the colors contains a monochromatic clique of size k. The case s=1 is the usual r-color Ramsey number, and the case s=r-1 was studied by Erdos, Hajnal and Rado in 1965, and by Erdos and Szemeredi in 1972. The first significant results for general s were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraete, who showed that R-r,R-s(k)=2(circle minus(kr)) if s/r is bounded away from 0 and 1. In the range s=r-o(r), however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine R-r,R-s(k) up to polylogarithmic factors in the exponent for essentially all r, s, and k.