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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
##### Keywords:
edge-coloring; projective planes; algebraic constructions
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