Small Ramsey numbers for books, wheels, and generalizations

TL;DR

This paper advances the understanding of Ramsey numbers for specific graph structures like books and wheels, providing new bounds and exact values using diverse computational and theoretical methods.

Contribution

It presents new upper and lower bounds, including exact values for certain Ramsey numbers involving wheels and books, employing multiple innovative techniques.

Findings

Established $R(W_5, W_7) = 15$

Proved $R(W_5, W_9) = 18$

Derived bounds for generalized Ramsey numbers $GR(r,K_s,t)$

Abstract

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$, and $R(B_3, B_7) = 20$, and a number of additional tight lower bounds for books. We use a range of different methods: flag algebras, local search, bottom-up generation, and enumeration of polycirculant graphs. We also explore generalized Ramsey numbers using similar methods. Let $GR(r,K_s,t)$ denote the minimum number of vertices $n$ such that any $r$-edge-coloring of $K_n$ has a copy of $K_s$ with at most $t$ colors. We establish $GR(3,K_4,2) = 10, GR(4,K_4,3) = 10$, and some additional bounds.