Size bipartite Ramsey numbers

Let BB, B1B1 and B2B2 be bipartite graphs, and let B→(B1,B2)B→(B1,B2) signify that any red–blue edge-coloring of BB contains either a red B1B1 or a blue B2B2. The size bipartite Ramsey number br̂(B1,B2) is defined as the minimum number of edges of a bipartite graph BB such that B→(B1,B2)B→(B1,B2). It is shown that br̂(Km,n,Km,n) is linear on nn with mm fixed, and br̂(Kn,n,Kn,n) is between c1n22nc1n22n and c2n32nc2n32n for some positive constants c1c1 and c2c2.