A lower bound for set-coloring Ramsey numbers

The set-coloring Ramsey number (Formula presented.) is defined to be the minimum (Formula presented.) such that if each edge of the complete graph (Formula presented.) is assigned a set of (Formula presented.) colors from (Formula presented.), then one of the colors contains a monochromatic clique of size (Formula presented.). The case (Formula presented.) is the usual (Formula presented.) -color Ramsey number, and the case (Formula presented.) was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general (Formula presented.) were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that (Formula presented.) if (Formula presented.) is bounded away from 0 and 1. In the range (Formula presented.), however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine (Formula presented.) up to polylogarithmic factors in the exponent for essentially all (Formula presented.), (Formula presented.), and (Formula presented.).