The Erdős-Gyárfás problem on generalized Ramsey numbers, submitted

Fix positive integers p and q with 2 ≤ q ≤ (p2). An edge coloring of the complete graph Kn is said to be a (p, q)-coloring if every Kp receives at least q different colors. The function f(n, p, q) is the minimum number of colors that are needed for Kn to have a (p, q)-coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f(n, p, p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n, p, q) is subpolynomial in n. We prove that the answer is p − 1.