Ramsey numbers of large books

A book Bn ${B}_{n}$ is a graph which consists of n $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies r(Bm,Bn)<= 2(m+n)+c $r({B}_{m},{B}_{n})\le \,2(m+n)+c$ for some constant c>0 $c\gt 0$. In this article, we obtain that r(Bm,Bn)<= 2(m+n)+o(n) $r({B}_{m},{B}_{n})\le 2(m+n)+o(n)$ for all m <= n $m\le n$ and n $n$ 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.