Another proof that the two color bipartite Ramsey number is $O(2^t)$

Abstract

For positive integers $t$ and $q$ let $b_q(t)$ be the smallest integer $n$ so that any coloring of the edges of the complete bipartite graph $K_{n,n}$ with $q$ colors yields a monochromatic copy of $K_{t,t}$. We give an independent proof that $b_2(t)\le 128\cdot 2^t$ for every positive integer $t$. More generally, for $0<p\le 1/2$, every bipartite graph of edge density at least $p$, with both classes of size at least $128\cdot p^{-t}$, contains $K_{t,t}$. This gives $b_q(t)\le 128\cdot q^t$ for every integer $q\ge 2$ and a uniform consequence for Zarankiewicz numbers.