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

TL;DR

This paper establishes new asymptotic bounds for the generalized Ramsey number $f(n,5,8)$, showing it grows linearly with $n$ within specific bounds, using advanced hypergraph matching techniques.

Contribution

The paper provides the first non-trivial bounds on $f(n,5,8)$, applying the conflict-free hypergraph matchings method to improve understanding of this generalized Ramsey number.

Findings

Lower bound: rac{6}{7}(n-1)

Upper bound: n + o(n)

Method: conflict-free hypergraph matchings

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 $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove $f(n,4,5) = \frac{5}{6}n + o(n)$.