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 ) ≤ 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 ) ≤ 2n + 3 for m ≤ n/6 − o(n) and large n. Thus the conjecture is true for m ≤ n/6 − o(n) and large n. In this paper, we obtain that r(B m , B n ) ≤ 2(m + n) + o(n) for n/6 ≤ m ≤ 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 ≤ α ≤ 1 and large n, r(This lower bound asymptotically matches the upper bound when α = 1 − o(1).