A new upper bound on the minimum degree of minimal Ramsey graphs

TL;DR

This paper establishes a new upper bound on the minimum degree of minimal Ramsey graphs for complete graphs, improving understanding of their structural properties in edge-coloring problems.

Contribution

It provides a novel upper bound on the minimum degree of minimal Ramsey graphs for complete graphs, advancing theoretical bounds in Ramsey theory.

Findings

Proves s_r(K_{k+1}) = O(k^3 r^3  k)

Improves bounds on minimal Ramsey graph degrees

Enhances understanding of edge-coloring constraints in Ramsey theory

Abstract

We prove that $s_r(K_{k+1}) = O(k^3 r^3 \log^3 k)$, 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.