Ramsey numbers of large books

A bookB n${B}_{n}$ is a graph which consists ofn $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfiesr ( B m , B n ) ≤ 2 ( m + n ) + c $r({B}_{m},{B}_{n})\le \,2(m+n)+c$ for some constantc > 0 $c\gt 0$ . In this article, we obtain thatr ( B m , B n ) ≤ 2 ( m + n ) + o ( n )$r({B}_{m},{B}_{n})\le 2(m+n)+o(n)$ for allm ≤ n $m\le n$ andn $n$ 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 on strongly regular graph holds asymptotically.