Sharp exponents for bipartite Erd\H{o}s-Rado numbers

The Erd\H{o}s-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 Erd\H{o}s-Rado number $ER_B(m)$ satisfies \[ \log ER_B(m) = \Theta(m \log m). \] 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 $\log m$.