Cutsets in ${\mathcal P}(X)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908489747529728 |
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| author | Ginsburg, John Sands, Bill |
| author_facet | Ginsburg, John Sands, Bill |
| contents | For any set $X$, ${\mathcal P}(X)$ denotes the collection of all subsets of $X$, ordered by inclusion. A {\it cutset} in ${\mathcal P}(X)$ is a subset of ${\mathcal P}(X)$ which meets every maximal chain of ${\mathcal P}(X)$. A cutset is non-trivial if it does not contain $X$ or the empty set. Our main result is the following.
Theorem 1: Let $X$ be an infinite set of cardinality $κ$. Every non-trivial cutset in ${\mathcal P}(X)$ contains a chain of cardinality $κ^+$ and an antichain of cardinality $2^κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cutsets in ${\mathcal P}(X)$ Ginsburg, John Sands, Bill Combinatorics Logic 03E04, 03E10, 06A06, 06D05, 06E05 For any set $X$, ${\mathcal P}(X)$ denotes the collection of all subsets of $X$, ordered by inclusion. A {\it cutset} in ${\mathcal P}(X)$ is a subset of ${\mathcal P}(X)$ which meets every maximal chain of ${\mathcal P}(X)$. A cutset is non-trivial if it does not contain $X$ or the empty set. Our main result is the following. Theorem 1: Let $X$ be an infinite set of cardinality $κ$. Every non-trivial cutset in ${\mathcal P}(X)$ contains a chain of cardinality $κ^+$ and an antichain of cardinality $2^κ$. |
| title | Cutsets in ${\mathcal P}(X)$ |
| topic | Combinatorics Logic 03E04, 03E10, 06A06, 06D05, 06E05 |
| url | https://arxiv.org/abs/2508.10221 |