The generalized Ramsey number $f(n,5,8)=\frac{6}{7}n+o(n)$

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 $f(n,5,8)\ge \frac{6}{7}(n-1)$. Here we prove an asymptotically matching upper bound.