K2,2–K1,n and K2,n–K2,n bipartite Ramsey numbers Original Research Article

Some bounds for G1–G2 bipartite Ramsey numbers b(G1;G2) are given, which imply that b(K2,2;K1,n)=n+q for the range q2−q+1⩽n⩽q2, where q is a prime power. Our new construction establishes in particular that b(K2,n;K2,n)=4n−3 if 4n−3 is a prime power, reinforcing a weaker form of a conjecture due to Beineke and Schwenk. Particular relationships between b(G1;G2) and G1–G2 Ramsey numbers are also determined.