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 2B₂- Our result implies that T_n is 2B₂-good if n ≥ 5. [ABSTRACT FROM AUTHOR]