The bipartite Ramsey numbers (Q540030)

Summary: Given bipartite graphs \(H_1\) and \(H_2\), the bipartite Ramsey number \(b(H_1; H_2)\) is the smallest integer \(b\) such that any subgraph \(G\) of the complete bipartite graph \(K_{b,b}\), either \(G\) contains a copy of \(H_1\) or its complement relative to \(K_{b,b}\) contains a copy of \(H_2\). It is known that \(b(K_{2,2}; K_{2,2}) = 5, b(K_{2,3}; K_{2,3}) = 9, b(K_{2,4}; K_{2,4}) = 14\) and \(b(K_{3,3}; K_{3,3}) = 17\). In this paper we study the case \(H_1\) being even cycles and \(H_2\) being \(K_{2,2}\), prove that \(b(C_6; K_{2,2}) = 5\) and \(b(C_{2m}; K_{2,2}) = m + 1\) for \(m \geq 4\).