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The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such thatevery blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or ared $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an emptygraph 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. Thisstep is repeated until no more edges can be added. In this note, we analyzethis process and show that the resulting graph witnesses that $b(2,t) =\Omega\left(t^{3/2}/\log t \right)$, thereby improving the best known lower bound.

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