The bipartite \(K_{2,2}\)-free process and bipartite Ramsey number \(b(2, t)\) (Q2209894)

Summary: The bipartite Ramsey number \(b(s,t)\) is the smallest integer \(n\) such that every blue-red edge coloring of \(K_{n,n}\) contains either a blue \(K_{s,s}\) or a red \(K_{t,t}\). In the bipartite \(K_{2,2}\)-free process, we begin with an empty graph on vertex set \(X\cup Y\), \(|X|=|Y|=n\). At each step, a random edge from \(X\times Y\) is added under the restriction that no \(K_{2,2}\) is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and prove that the resulting graph shows that \(b(2,t) =\Omega(t^{3/2}/\log t)\), thereby improving the best known lower bound.