Ramsey numbers of large books

TL;DR

This paper proves an asymptotic upper bound for the Ramsey numbers of large books, confirming a longstanding conjecture and implications for related graph theory conjectures.

Contribution

It establishes an asymptotic bound for Ramsey numbers of large books, confirming Rousseau and Sheehan's conjecture.

Findings

Proves $r(B_m, B_n) o 2(m+n)$ as $n o

Confirms Rousseau and Sheehan's conjecture asymptotically

Implications for a conjecture on strongly regular graphs

Abstract

A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies $r(B_m,B_n)\le 2(m+n)+c$ for some constant $c>0$. In this paper, we obtain that $r(B_m, B_n)\le 2(m+n)+o(n)$ for all $m\le n$ and $n$ large, which confirms the conjecture of Rousseau and Sheehan asymptotically. As a corollary, our result implies that a related conjecture of Faudree, Rousseau and Sheehan (1982) on strongly regular graph holds asymptotically.