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arXiv · 2410.08982

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

Dániel Dobák·Eion Mulrenin
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Abstract

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 $\log m$.

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Dániel Dobák, Eion Mulrenin. 2024-10-11. Sharp exponents for bipartite Erd\H{o}s-Rado numbers. https://arxiv.org/abs/2410.08982

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