The 5-Bipartite Ramsey Numbers of Graphs K_(2,3) and K_(3,3)

摘要

For a positive integer s and two bipartite graphs F and H, the s-bipartite Ramsey number BR_s(F,H) of F and H is the smallest integer t with t ≥ s such that every red-blue coloring of K_(s,t) results in a red F or a blue H. In this paper, the numbers BR.,(K_(2,3),k_(3,3)) are investigated for several values of s. Specifically, it is shown that (i) BR_s(K_(2,3),K_(3,3)) = 21 for s = 4,5, (ii) BR_s(K_(2,3), K_(3,3)) = 15 for s = 6,7 and (iii) BR_s(K_(2,3), K_(3,3)) € {13,14} for s = 8,9. It is also shown that BR_s(K_(2,3), K_(3,3)) ≤ 14 for all s with 8 ≤ s ≤ BR(K_(2,3),K_(3,3)).