Some Fan-wheel Ramsey Numbers

Since1928, the British mathematician Ramsey defined Ramsey num-ber，Graham, Rothschild and Spencer have developed Ramsey number into Ramsey Theory. The theory describes that for any sufficiently large discrete system there must be a specific subsystem of certain structure. In recent years, we pay more and more attention to the theory.In this paper, we study the Ramsey number of some graphs. For two given graphs G1and G2, the Ramsey number R(G1, G2) is the smallest integer n satisfying that for every graph G of order n either G contains G1or its complement G contains G2.Let Fn, Wn. Cn be a fan of order2n+1, a wheel of n+1and a cycle of n, respectively. In the context [Journal of Graph Theory,7(1983),39-51], S.A. Burr and P. Erdos proved that R(F1Wn)=R(C3, Wn)=2n+1. This paper mainly studies the Ramsey number of wheels versus a fan of order five, and it is proved that if n≥9, then R(F2, Wn)=2n+1. The first chapter mainly describes the development of Ramsey theory and introduces the basic concepts of graph theory and the notation used in this paper. The second chapter tells the latest progress and some results about classical Ramsey number, the Ramsey number of graphs and the Ramsey number about fans which we will study in this article. The third chapter proves the main result about the Ramsey number of F2versus wheels:R(F2, Wn)=2n+1.