Bipartite Ramsey numbers involving large

Let b r ( H 1 , H 2 ) be the bipartite Ramsey number for bipartite graphs H 1 and H 2 . It is shown that the order of magnitude of b r ( K t , n , K n , n ) is n t + 1 / ( log n ) t for t ≥ 1 fixed and n → ∞ . Moreover, if H is an isolate-free bipartite graph of order h having bipartition ( A , B ) that satisfies Δ ( B ) ≤ t , then b r ( H , K n , n ) can be bounded from above by ( h n / log n ) t ( log n ) α ( t ) for large n , where α ( 1 ) = α ( 2 ) = 1 and α ( t ) = 0 for t ≥ 3 .