Extensions of a Family for Sunflowers

TL;DR

This paper extends the understanding of sunflower structures in combinatorial set families, establishing new conditions under which such families contain disjoint sets or sunflowers, based on the $ ext{Ext}$ operation and $ ext{Gamma}$-conditions.

Contribution

It introduces refined $ ext{Gamma}$-conditions that guarantee the existence of disjoint sets and sunflowers in set families, advancing combinatorial sunflower theory.

Findings

Family $ ext{Ext}$-properties relate to sunflower existence.

New $ ext{Gamma}$-conditions ensure disjoint sets and sunflowers.

Results improve bounds for sunflower and disjoint set inclusion.

Abstract

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 $\Gamma(b)$-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 $b= m^{\frac{1}{2}+ \epsilon}$ with an $m$ sufficiently larger than a given constant $1/\epsilon$. 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 $\Gamma \left( 8^{\sqrt{\log_2 m}} \sqrt m ~k \log_2 k \right)$-condition with an $m$ sufficiently larger than $k$.