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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2504.13347 |
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| _version_ | 1866912687106031616 |
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| author | Gendler, Gabriel |
| author_facet | Gendler, Gabriel |
| contents | The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can pose more general conjectures by choosing a different probability distribution on the cube. In particular, for any sequence of probabilities $(p_i)_{i=1}^d$ we can consider the product of $d$ independent Bernoulli random variables, with success probabilities $p_i$. In this short note, we find a generalised form of Karpas' special case of the union-closed conjecture for families $\mathcal{F}$ with density at least half. We also generalise Knill's logarithmic lower bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partial results for union-closed conjectures on the weighted cube Gendler, Gabriel Combinatorics 97K20 The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can pose more general conjectures by choosing a different probability distribution on the cube. In particular, for any sequence of probabilities $(p_i)_{i=1}^d$ we can consider the product of $d$ independent Bernoulli random variables, with success probabilities $p_i$. In this short note, we find a generalised form of Karpas' special case of the union-closed conjecture for families $\mathcal{F}$ with density at least half. We also generalise Knill's logarithmic lower bound. |
| title | Partial results for union-closed conjectures on the weighted cube |
| topic | Combinatorics 97K20 |
| url | https://arxiv.org/abs/2504.13347 |