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On generalized Ramsey theory: The bipartite case. (English) Zbl 1023.05101
Summary: Given graphs \(G\) and \(H\), a coloring of \(E(G)\) is called an \((H,q)\)-coloring if the edges of every copy of \(H\subseteq G\) 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 \(r(K_{n,n}, K_{p,p},q)\) is linear in \(n\), the smallest \(q\) for which it is quadratic in \(n\). We also determine the smallest \(q\) for which \(r(K_{n,n}, K_{p,p},q)= n^2- O(n)\), and the smallest \(q\) for which \(r(K_{n,n}, K_{p,p},q)= n^2- O(1)\). Our results include showing that \(r(K_{n,n}, K_{2,t+1},2)\) and \(r(K_n, K_{2,t+1}, 2)\) are both \((1+ o(1))\sqrt{n/t}\) as \(n\to\infty\), thereby proving a special case of a conjecture of F. R. K. Chung and R. L. Graham [J. Comb. Theory, Ser. B 18, 164-169 (1975; Zbl 0298.05122)]. Finally, we determine the exact value of \(r(K_{n,n}, K_{3,3},8)\), and prove that \(2n/3\leq r(K_{n,n}, C_4,3)\leq n+1\). Several problems remain open.

MSC:
05C55 Generalized Ramsey theory
05D10 Ramsey theory
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