The generalized Ramsey number $f(n, 5, 8) = \frac 67 n + o(n)$

TL;DR

This paper determines the asymptotic value of the generalized Ramsey number f(n, 5, 8), showing it is approximately 6/7 of n, by establishing matching bounds.

Contribution

The authors prove an asymptotically tight upper bound for the generalized Ramsey number f(n, 5, 8), complementing previous lower bounds.

Findings

Established that f(n, 5, 8) = (6/7)n + o(n) asymptotically.

Provided a matching upper bound to previous lower bounds.

Contributed to understanding edge colorings in complete graphs with clique and color constraints.

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.