A New View of Bipartite Ramsey Numbers

For a bipartite graph \( G \) and a positive integer \( s \), the \( s \)-bipartite Ramsey number \( BR_s(G) \) of \( G \) is the minimum integer \( t \) with \( t \geq s \) for which every red-blue coloring of \( K_{s,t} \) results in a monochromatic \( G \). A formula is established for \( BR_s(K_{r,r}) \) when \( s \in \{2r – 1, 2r\} \) when \( r \geq 2 \). The \( s \)-bipartite Ramsey numbers are studied for \( K_{3,3} \) and various values of \( s \). In particular, it is shown that \( BR_5(K_{3,3}) = 41 \) when \( s \in \{5,6\} \), \( BR_s(K_{3,3}) = 29 \) when \( s \in \{7,8\} \), and \( 17 \leq BR_{10}(K_{3,3}) \leq 23 \).