Size bipartite Ramsey numbers

Let B , B 1 and B 2 be bipartite graphs, and let B → ( B 1 , B 2 ) signify that any red–blue edge-coloring of B contains either a red B 1 or a blue B 2 . The size bipartite Ramsey number b r ̂ ( B 1 , B 2 ) is defined as the minimum number of edges of a bipartite graph B such that B → ( B 1 , B 2 ) . It is shown that b r ̂ ( K m , n , K m , n ) is linear on n with m fixed, and b r ̂ ( K n , n , K n , n ) is between c 1 n 2 2 n and c 2 n 3 2 n for some positive constants c 1 and c 2 .