Ramsey numbers of large books
Abstract
A book Bn is a graph which consists of n triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies r(Bm,Bn) 2(m+n)+c for some constant c>0. In this paper, we obtain that r(Bm, Bn) 2(m+n)+o(n) for all m n and 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 (1982) on strongly regular graph holds asymptotically.
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