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 ∨ K n ¯ . Erdős, Faudree, Rousseau and Schelp proved that T n is B m -good if n ≥ 3 m − 3 . In this paper, we obtain the exact Ramsey number of T n versus 2 B 2 - Our result implies that T n is 2 B 2 -good if n ≥ 5.