Upper Bounds for Two Bipartite Ramsey Numbers

Abstract:

It is shown that br(Ts,Kn,n)≤sn for all integers n≥s≥3045, and br(Kt,t,Km,n)≤n+cn1-1/t for fixed integers t≥2,m≥1,where c>0 is a constant.Moreover,br(Kt,t,Km,n-m) for positive integers is obtained,which improves the lower bound r(Kt,t,Km,n-m)/2.

Key words:graph theory; bipartite ramsey number; Zarankiewicz number