On the number of high-dimensional partitions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866913244861431808 |
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| author | Pohoata, Cosmin Zakharov, Dmitriy |
| author_facet | Pohoata, Cosmin Zakharov, Dmitriy |
| contents | Let $P_{d}(n)$ denote the number of $n \times \ldots \times n$ $d$-dimensional partitions with entries from $\left\{0,1,\ldots,n\right\}$. Building upon the works of Balogh-Treglown-Wagner and Noel-Scott-Sudakov, we show that when $d \to \infty$, $$P_{d}(n) = 2^{(1+o_{d}(1)) \sqrt{\frac{6}{(d+1)π}} \cdot n^{d}}$$ holds for all $n \geq 1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_02436 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the number of high-dimensional partitions Pohoata, Cosmin Zakharov, Dmitriy Combinatorics Number Theory Let $P_{d}(n)$ denote the number of $n \times \ldots \times n$ $d$-dimensional partitions with entries from $\left\{0,1,\ldots,n\right\}$. Building upon the works of Balogh-Treglown-Wagner and Noel-Scott-Sudakov, we show that when $d \to \infty$, $$P_{d}(n) = 2^{(1+o_{d}(1)) \sqrt{\frac{6}{(d+1)π}} \cdot n^{d}}$$ holds for all $n \geq 1$. 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. |
| title | On the number of high-dimensional partitions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2112.02436 |