  A book $B_n$ is a graph consisting of $n$ triangles sharing a common edge. In
1978, Rousseau and Sheehan conjectured that the Ramsey number $r(B_m,B_n)\le
2(m+n+1)+c$ for some constant $c>0$. In particular, Nikiforov and Rousseau
(Book Ramsey numbers I, Random Structures Algorithms 27 (2005), 379--400)
proved that $r(B_m,B_n)\le2n+3$ for $m\le n/6-o(n)$ and large $n$. Thus the
conjecture is true for $m\le n/6-o(n)$ and large $n$. In this paper, we obtain
that $r(B_m, B_n)\le 2(m+n)+o(n)$ for $n/6\le m\le n$ and large $n$, so the
above conjecture holds asymptotically for the remaining case. Our result also
implies that a conjecture of Faudree, Rousseau and Sheehan (1982) on strongly
regular graph holds asymptotically. Moreover, we prove that for
$1/6\le\alpha\le1$ and large $n$, $r(B_{\lceil\alpha n\rceil}, B_n)\ge
(\sqrt{4\alpha}+1+\alpha-o(1))n$. This lower bound asymptotically matches the
upper bound when $\alpha=1-o(1)$.
