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The generalized Ramsey number $f(n, 5, 8) = \frac 67 n + o(n)$
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abstract
A $(p, q)$-coloring of $K_n$ is a coloring of the edges of $K_n$ such that every $p$-clique has at least $q$ distinct colors among its edges. The generalized Ramsey number $f(n, p, q)$ is the minimum number of colors such that $K_n$ has a $(p, q)$-coloring. Gomez-Leos, Heath, Parker, Schweider and Zerbib recently proved $f(n, 5, 8) \ge \frac 67 (n-1)$. Here we prove an asymptotically matching upper bound.
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Edge-coloring $K_{n, n}$ with no 2-colored $C_{2k}$
The minimum number of colors in a (C_{2k},3)-coloring of K_{n,n} is exactly (7/20)n+o(n) for k=3 and lies between improved explicit bounds for all k≥4.
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