A lower bound for set-coloring Ramsey numbers

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