New upper bounds for Ramsey numbers of books

Abstract

A book Bn is a graph which consists of n triangles sharing a common edge. Rousseau and Sheehan (1978) conjectured that r(Bm,Bn)≤2(m+n+1)+c some constant c>0. Let m=⌊αn⌋ where 0<α≤1 is a real number. A result of Nikiforov and Rousseau [Random Structures Algorithms 27 (2005), 379–400] implies that this conjecture holds in a stronger form for 0<α≤1/6 and large n. We prove that r(Bm,Bn)≤(3/2+3α+o(1))n, where 1/4<α<1/2. This confirms the conjecture in a stronger form for 1/6≤α<1/2 and large n. As a corollary, r(B⌈n4⌉,Bn)=(9/4+o(1))n.