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arXiv · 1808.02139

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

Deepak Bal·Patrick Bennett
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Abstract

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 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.

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Deepak Bal, Patrick Bennett. 2018-08-06. The bipartite $K_{2,2}$-free process and bipartite Ramsey number $b(2, t)$. https://arxiv.org/abs/1808.02139

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