Abstract
In this paper we show that r(C4,Kt,t)≥Ω(t3∕2logt) via quasi-random graphs giving a polylogarithmic improvement over the currently best lower bound, which implies r(C4,Kt)≥Ω(t3∕2logt) and br(C4,Kt,t)≥Ω(t3∕2logt), where br(C4,Kt,t) is the bipartite Ramsey number of C4 and Kt,t. This builds on a recent breakthrough of Mubayi and Verstraëte (2019) reducing off-diagonal Ramsey numbers to the existence of certain quasi-random graphs.
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