Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Asymptotic Bounds for Bipartite Ramsey Numbers. | Asymptotic Bounds for Bipartite Ramsey Numbers Yair Caro Department of Mathematics University of Haifa - Oranim Tivon 36006 Israel ya_caro@ Cecil Rousseau Department of Mathematical Sciences The University of Memphis Memphis TN 38152-3240 ccrousse@ Submitted July 11 2000 Accepted February 7 2001. MR Subject Classifications 05C55 05C35 Abstract The bipartite Ramsey number b m n is the smallest positive integer r such that every red green coloring of the edges of Kr r contains either a red Km m or a green Kn n. We obtain asymptotic bounds for b m n for m 2 fixed and n 1. 1 Introduction Recent exact results for bipartite Ramsey numbers 4 have rekindled interest in this subject. The bipartite Ramsey number b m n is the smallest integer r such that every red green coloring of the edges of Kr r contains either a red Km m or a green Kn n. In early work on the subject 1 Beineke and Schwenk proved that b 2 2 5 and b 3 3 17. In 4 Hattingh and Henning prove that b 2 3 9 and b 2 4 14. The following variation was considered by Beineke and Schwenk 1 and also by Irving 5 for 1 m n the bipartite Ramsey number R m n is the smallest integer r such that every red green coloring of the edges of Kr r contains a monochromatic Km n. Irving found that R 2 n 4n 3 with equality if n is odd and there is Hadamard matrix of order 2 n 1 . The bound R m n 2m n 1 1 was proved by Thomason in 7 . Note that b m m R m m . In this note we obtain asymptotic bounds for b m n with m fixed and n 00. 2 The Main Result Theorem 1. Let m 2 be fixed. Then there are constants A and B such that m 1 2 m a - b m n - i n 1. log n log n THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R17 1 Specifically these bounds hold with A 1 e m 1 m m2 and B 1 m1 m 1 where e 0 is arbitrary. Proof. The upper bound is based on well-known results for the Zarankiewicz function. Let z r s denote the maximum number of edges that a subgraph of Krr can have if it does not contain KS S as a subgraph. We use the .
