Ramsey with purple edges

TL;DR

This paper explores a variant of Ramsey numbers involving edges that are simultaneously red and blue, called purple, and determines the maximum number of purple edges in such colorings without certain monochromatic cliques.

Contribution

It introduces and asymptotically determines the maximum purple edges in a new Ramsey variant involving purple edges, revealing dependencies with Ramsey-Turán numbers.

Findings

Asymptotic determination of maximum purple edges for various parameters

Establishment of strong links between purple-edge Ramsey variants and Ramsey-Turán numbers

Extension of classical Ramsey theory to multi-color and multi-edge scenarios

Abstract

Motivated by a question of Angell, we investigate a variant of Ramsey numbers where some edges are coloured simultaneously red and blue, which we call purple. Specifically, we are interested in the largest number $g=g(n;s,t)$, for some $s$ and $t$ and $n<R(s,t)$, such that there exists a red/blue/purple colouring of $K_n$ with $g$ purple edges, with no red/purple copy of $K_s$ nor blue/purple copy of $K_t$. We determine $g$ asymptotically for a large family of parameters, exhibiting strong dependencies with Ramsey-Tur\'{a}n numbers.