Further analysis on the second frequency of union-closed set families
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912147672399872 |
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| author | Wu, Saintan |
| author_facet | Wu, Saintan |
| contents | The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\frac{1}{2^{k-1}+1}$ proportion of the sets in the family.
Das and Wu showed that this conjecture is true for $k\geq 3$ and $k=2$ if $|\mathcal{F}|$ is outside a particular range. In this companion paper, we analyse further when $\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Further analysis on the second frequency of union-closed set families Wu, Saintan Combinatorics 05D05 The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\frac{1}{2^{k-1}+1}$ proportion of the sets in the family. Das and Wu showed that this conjecture is true for $k\geq 3$ and $k=2$ if $|\mathcal{F}|$ is outside a particular range. In this companion paper, we analyse further when $\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming. |
| title | Further analysis on the second frequency of union-closed set families |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2412.03863 |