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

TL;DR

This paper analyzes a bipartite process avoiding $K_{2,2}$ subgraphs, establishing a new lower bound for bipartite Ramsey numbers $b(2,t)$ that improves previous estimates.

Contribution

It introduces and analyzes a bipartite $K_{2,2}$-free process, deriving a sharper lower bound for bipartite Ramsey numbers $b(2,t)$ compared to prior work.

Findings

Established that $b(2,t) = ext{Omega}(t^{3/2}/ ext{log } t)$

Improved the known lower bounds for bipartite Ramsey numbers

Provided insights into the structure of bipartite graphs avoiding $K_{2,2}$

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.