Sharp Exponents for Bipartite Erdős‐Rado Numbers

ABSTRACT The Erdős–Rado canonization theorem generalizes Ramsey's theorem to edge‐colorings with an unbounded number of colors, in the sense that for sufficiently large, any edge‐coloring of will yield some copy of 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 satisfies in contrast with the non‐bipartite setting where the best known lower and upper bounds on are still separated by a factor of .