On the number of high-dimensional partitions

Abstract

Let $P_d(n)$ denote the number of $n\times \dots \times n$ $d$-dimensional partitions with entries from $\left\{0,1,\dots ,n\right\}$. Building upon the works of Balogh-Treglown-Wagner and Noel-Scott-Sudakov, we show that when $d\to \infty$, $$P_d(n)=2^{(1+o_d(1))\sqrt{\frac{6}{(d+1)\pi }}\cdot n^d}$$ holds for all $n\ge 1$. This makes progress towards a conjecture of Moshkovitz-Shapira [{\it{Adv. in Math.}} 262 (2014), 1107--1129]. Via the main result of Moshkovitz and Shapira, our estimate also determines asymptotically a Ramsey theoretic parameter related to Erdős-Szekeres-type functions, thus solving a problem of Fox, Pach, Sudakov, and Suk [{\it{Proc. Lond. Math. Soc.}} 105 (2012), 953--982]. Our main result is a new supersaturation theorem for antichains in $[n{]}^d$, which may be of independent interest.

References in corpus (1)

• Independent sets in the hypercube revisited

Cited by in corpus (1)

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