Ramsey Numbers of Trees Versus Multiple Copies of Books

Given two graphs G and H,the Ramsey number R(G,H) is the minimum integer N such that any two-coloring of the edges of K_(N) in red or blue yields a red G or a blue H.Let v(G) be the number of vertices of G and χ(G) be the chromatic number of G.Let s(G) denote the chromatic surplus of G,the number of vertices in a minimum color class among all proper χ(G)-colorings of G.Burr showed that R(G,H)≥(v(G)-1)(χ(H)-1)+s(H) if G is connected and v(G)≥s(H).A connected graph G is H-good if R(G,H)=(v(G)-1)(χ(H)-1)+s(H).Let tH denote the disjoint union of t copies of graph H,and let G∨H denote the join of G and H.Denote a complete graph on n vertices by K_(n),and a tree on n vertices by T_(n).Denote a book with n pages by B_(n),i.e.,the join K_(2) ∨■.Erd?s,Faudree,Rousseau and Schelp proved that T_(n) is B_(m)-good if n≥3m-3.In this paper,we obtain the exact Ramsey number of T_(n) versus 2B_(2).Our result implies that T_(n) is 2B_(2)-good if n≥5.