Sharp exponents for bipartite Erdős-Rado numbers

The Erdős-Rado canonization theorem generalizes Ramsey’s theorem to edge-colorings with an unbounded number of colors, in the sense that for $n=ER(m)$ sufficiently large, any edge-coloring of $E(K_n)\to \mathbb{N}$ will yield some copy of $K_m$ which is colored according to one of four canonical patterns. In this paper, we show that in the bipartite setting, the bipartite Erdős-Rado number $E{R}_B(m)$ satisfies $$\log E{R}_B(m)=\mathrm{\Theta }(m\log m)\mathrm{.}$$ Comparing this to the non-bipartite setting, the best known lower and upper bounds on $\log ER(m)$ are still separated by a factor of $\log m$.