The minimum degree of minimal Ramsey graphs for cliques

TL;DR

This paper establishes an upper bound on the minimum degree of minimal Ramsey graphs for cliques using a group-theoretic construction, advancing understanding of the structural properties of such graphs.

Contribution

It provides a new upper bound on the Ramsey parameter for cliques, employing a novel group-theoretic approach with generalized quadrangles.

Findings

Proves that s_r(K_k) = O(k^5 r^{5/2})

Uses a group-theoretic model of generalized quadrangles

Advances bounds on minimal Ramsey graphs for cliques

Abstract

We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.