New Lower Bounds for 28 Classical Ramsey Numbers

TL;DR

This paper presents new lower bounds for 28 classical Ramsey numbers using heuristic search and innovative colorings derived from known constructions, advancing the understanding of Ramsey theory.

Contribution

It introduces novel lower bounds for 28 Ramsey numbers by employing heuristic search and constructing new colorings from existing configurations.

Findings

New lower bounds for 28 Ramsey numbers established.

Use of $(5,k)$- and $(7,k)$-colorings to derive new bounds.

Application of Paley and cubic colorings in constructions.

Abstract

We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures we used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well-known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$.