Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.04219 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916710816153600 |
|---|---|
| author | Fukuyama, Junichiro |
| author_facet | Fukuyama, Junichiro |
| contents | This paper explores the structure of the combinatorial domain $2^X$ in relation to sunflowers. The previous study found some intrinsic properties of the $l$-extension \[ Ext \left( \mathcal{F}, l \right) = \left\{ V ~:~ V \in {X \choose l},~ \exists U \in \mathcal{F}~ U \subset V \right\} \] of a family $\mathcal{F}$ of $m$-cardinality sets. Subsequently, it lead to the proof that such an $\mathcal{F}$ includes three mutually disjoint sets if it satisfies the $Γ(b)$-condition, that is, \[ \left| \mathcal{F}[S] \right| < b^{-|S|} |\mathcal{F}| \quad \textrm{for every nonempty set}~ S, \qquad \textrm{where} \quad \mathcal{F} [S] := \left\{ U : U \in \mathcal{F},~ S \subset U \right\}, \] for $b= m^{\frac{1}{2}+ ε}$ with an $m$ sufficiently larger than a given constant $1/ε$. It is stronger than the statement that $\mathcal{F}$ includes a 3-sunflower if $|\mathcal{F}| > b^m$, where $k$-sunflower refers to a family of $k$ different sets with a common pair-wise intersection. Further refining the theory, we show that an $\mathcal{F}$ includes $k$ mutually disjoint sets if it satisfies the $Γ\left( 8^{\sqrt{\log_2 m}} \sqrt m ~k \log_2 k \right)$-condition with an $m$ sufficiently larger than $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04219 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extensions of a Family for Sunflowers Fukuyama, Junichiro Combinatorics 05D05 This paper explores the structure of the combinatorial domain $2^X$ in relation to sunflowers. The previous study found some intrinsic properties of the $l$-extension \[ Ext \left( \mathcal{F}, l \right) = \left\{ V ~:~ V \in {X \choose l},~ \exists U \in \mathcal{F}~ U \subset V \right\} \] of a family $\mathcal{F}$ of $m$-cardinality sets. Subsequently, it lead to the proof that such an $\mathcal{F}$ includes three mutually disjoint sets if it satisfies the $Γ(b)$-condition, that is, \[ \left| \mathcal{F}[S] \right| < b^{-|S|} |\mathcal{F}| \quad \textrm{for every nonempty set}~ S, \qquad \textrm{where} \quad \mathcal{F} [S] := \left\{ U : U \in \mathcal{F},~ S \subset U \right\}, \] for $b= m^{\frac{1}{2}+ ε}$ with an $m$ sufficiently larger than a given constant $1/ε$. It is stronger than the statement that $\mathcal{F}$ includes a 3-sunflower if $|\mathcal{F}| > b^m$, where $k$-sunflower refers to a family of $k$ different sets with a common pair-wise intersection. Further refining the theory, we show that an $\mathcal{F}$ includes $k$ mutually disjoint sets if it satisfies the $Γ\left( 8^{\sqrt{\log_2 m}} \sqrt m ~k \log_2 k \right)$-condition with an $m$ sufficiently larger than $k$. |
| title | Extensions of a Family for Sunflowers |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2301.04219 |