New bounds on the generalized Ramsey number $f(n,5,8)$

Abstract

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1)\le f(n,5,8)\le n+o(n)$. The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove $f(n,4,5)=\frac{5}{6}n+o(n)$.