On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929600119963648 |
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| author | Jenssen, Matthew Malekshahian, Alexandru Park, Jinyoung |
| author_facet | Jenssen, Matthew Malekshahian, Alexandru Park, Jinyoung |
| contents | Dedekind's problem, dating back to 1897, asks for the total number $ψ(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $ψ(n)$ and the number of antichains of size $β\binom{n}{\lfloor n/2 \rfloor}$ for any fixed $β>0$. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice Jenssen, Matthew Malekshahian, Alexandru Park, Jinyoung Combinatorics Dedekind's problem, dating back to 1897, asks for the total number $ψ(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $ψ(n)$ and the number of antichains of size $β\binom{n}{\lfloor n/2 \rfloor}$ for any fixed $β>0$. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$. |
| title | On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.03400 |