On s-Bipartite Ramsey Numbers Stars, Matchings and Double Stars

For bipartite graphs \( F \) and \( H \) and a positive integer \( s \), the \( s \)-bipartite Ramsey number \( BR_s(F,H) \) of \( F \) and \( H \) is the smallest integer \( t \) with \( t \geq s \) such that every red-blue coloring of \( K_{s,t} \) results in a red \( F \) or a blue \( H \). We evaluate this number for all positive integers \( s \) when \( F \) and \( H \) are both stars, are both matchings, or one is a star and the other is a matching, as well as when \( F = H \) is an arbitrary double star.