Graphs and Combinatorics DOI 10.1007/s00373-013-1339-3 ORIGINAL PAPER Some Bistar Bipartite Ramsey Numbers Johannes H. Hattingh · Ernst J. Joubert Received: 7 June 2012 / Revised: 10 May 2013 © Springer Japan 2013 Abstract For bipartite graphs G 1 , G 2 ,..., G k , the bipartite Ramsey number b(G 1 , G 2 ,..., G k ) is the least positive integer b so that any colouring of the edges of K b,b with k colours will result in a copy of G i in the i th colour for some i . A tree of diameter three is called a bistar, and will be denoted by B(s , t ), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b( B(s , t ), B(s , t )) and b( B(s , s ), B(s , s )). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then b k ( B(s , s )) ≤⌈k (s - 1) +  (s - 1) 2 (k 2 - k ) - k (2s - 4)⌉. Finally, we show that for s ≥ 3 and k ≥ 2, the Ram- sey number r k ( B(s , s )) ≤⌈2k (s - 1) + 1 2 + 1 2  (4k (s - 1) + 1) 2 - 8k (2s 2 - s - 2)⌉. Keywords Bipartite graph · Ramsey · Bistar 1 Introduction For bipartite graphs G 1 , G 2 ,..., G k , the bipartite Ramsey number b(G 1 , G 2 ,..., G k ) is the least positive integer b so that any colouring of the edges of K b,b with k colours will result in a copy of G i in the i th colour for some i . The existence of all numbers b(G 1 , G 2 ,..., G k ) follows from a result of Erdös and Rado [5]. Similarly, the Ramsey number r (G 1 , G 2 ,..., G k ) is the least positive integer b so that any colouring of the J. H. Hattingh (B ) Department of Mathematics, East Carolina University, Greenville, NC 27858, USA e-mail: hattinghj@ecu.edu J. H. Hattingh · E. J. Joubert Department of Mathematics, University of Johannesburg, Auckland Park, South Africa e-mail: ejoubert@uj.ac.za 123