On Generalized Ramsey Theory: The Bipartite Case

Given graphs G and H, a coloring of E(G) is called an (H, q)-coloring if the edges of every copy of H⊆G together receive at least q colors. Let r(G, H, q) denote the minimum number of colors in an (H, q)-coloring of G. We determine, for fixed p, the smallest q for which r(Kn, n, Kp, p, q) is linear in n, the smallest q for which it is quadratic in n. We also determine the smallest q for which r(Kn, n, Kp, p, q)=n2−O(n), and the smallest q for which r(Kn, n, Kp, p, q)=n2−O(1). Our results include showing that r(Kn, n, K2, t+1, 2) and r(Kn, K2, t+1, 2) are both (1+o(1)) n/t as n→∞, thereby proving a special case of a conjecture of Chung and Graham. Finally, we determine the exact value of r(Kn, n, K3, 3, 8), and prove that 2n/3⩽r(Kn, n, C4, 3)⩽n+1. Several problems remain open.