New bounds for Ramsey numbers $R(K_k-e,K_l-e)$

TL;DR

This paper introduces algorithms to enumerate specific Ramsey graphs, leading to new bounds and exact values for various Ramsey numbers, including some previously unknown, and proves the uniqueness of a particular extremal graph.

Contribution

The authors develop algorithms for enumerating circulant and block-circulant Ramsey graphs, establishing new lower bounds, exact values, and the uniqueness of certain extremal graphs for multiple Ramsey numbers.

Findings

Proved $R(J_5,J_6)=37$ and $R(J_5,J_7)=65.

Established new lower bounds for several Ramsey numbers.

Identified the unique extremal graph for $R(J_5,J_7)$.

Abstract

Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let $J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: $49 \leq R(K_3,J_{12})$, $36 \leq R(J_4,K_8)$, $43 \leq R(J_4,J_{10})$, $52 \leq R(K_4,J_8)$, $37 \leq R(J_5,J_6)$, $43 \leq R(J_5,K_6)$, $65\leq R(J_5,J_7)$. We also use a gluing strategy to derive a new upper bound on $R(J_5,J_6)$. With both strategies combined, we prove the value of two Ramsey numbers: $R(J_5,J_6)=37$ and $R(J_5,J_7)=65$. We also show that the 64-vertex extremal Ramsey graph for $R(J_5,J_7)$ is unique. Furthermore, our algorithms also allow to establish new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: $R(W_7,W_4) = 21$, $R(W_7,W_7) = 19$, $R(K_{3,4},K_{3,4}) = 25$, and $R(K_{3,5}, K_{3,5})=33$.