On Bipartite Generalized Ramsey Theory

Abstract

Given graphs \(G\) and \(H\), an edge coloring of \(G\) is called an \((H,q)\)-coloring if the edges of every copy of \(H \subset G\) 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 [6] Erdős and Gyárfás studied \(r(K_n,K_p,q)\) if \(p\) and \(q\) are fixed and \(n\) tends to infinity. They determined for every fixed \(p\) the smallest \(q\) for which \(r(K_n,K_p,q)\) is linear in \(n\) and the smallest \(q\) for which \(r(K_n,K_p,q)\) is quadratic in \(n\). In [9] we studied what happens between the linear and quadratic orders of magnitude. In [2] Axenovich, Füredi, and Mubayi generalized some of the results of [6] to \(r(K_{n,n},K_{p,p},q)\). In this paper, we adapt our results from [9] to the bipartite case, namely we study \(r(K_{n,n},K_p,p,q)\) between the linear and quadratic orders of magnitude. In particular, we show that we can have at most \(\log p + 1\) values of \(q\) which give a linear \(r(K_{n,n},K_{p,p},q)\).