On hierarchically closed fractional intersecting families
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914748436578304 |
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| author | Balachandran, Niranjan Bhattacharya, Srimanta Kher, Krishn Vishwas Mathew, Rogers Sankarnarayanan, Brahadeesh |
| author_facet | Balachandran, Niranjan Bhattacharya, Srimanta Kher, Krishn Vishwas Mathew, Rogers Sankarnarayanan, Brahadeesh |
| contents | For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $θ\in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ θ\lvert A_1 \rvert, \dotsc, θ\lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{θ\}$ any fractional $r$-closed $θ$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $θ= 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02540 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On hierarchically closed fractional intersecting families Balachandran, Niranjan Bhattacharya, Srimanta Kher, Krishn Vishwas Mathew, Rogers Sankarnarayanan, Brahadeesh Combinatorics Primary 05D05, Secondary 05B99, 03E05 For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $θ\in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ θ\lvert A_1 \rvert, \dotsc, θ\lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{θ\}$ any fractional $r$-closed $θ$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $θ= 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism. |
| title | On hierarchically closed fractional intersecting families |
| topic | Combinatorics Primary 05D05, Secondary 05B99, 03E05 |
| url | https://arxiv.org/abs/2211.02540 |