An application of the regularity lemma in generalized Ramsey theory

Given graphs G and H, an edge coloring of 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 a (H,q )coloring of G. In [9] Erdős and Gyárfás studied r (Kn,Kp,q ) if p and q are fixed and n tends to infinity. They determined for every fixed p the smallest q (denoted by qlin) for which r (Kn,Kp,q ) is linear in n and the smallest q (denoted by qquad ) for which r (Kn,Kp,q ) is quadratic in n. They raised the problem of determining the smallest q for which we have r (Kn,Kp,q ) 1⁄4 (2) 0 (n). In this paper by using the Regularity Lemma we show that if q > qquad þ d2 2 e, then we have r (Kn,Kp,q ) 1⁄4 ( n 2) 0 (n). 2003 Wiley Periodicals, Inc. J Graph Theory 44: 39–49, 2003