Extensions of a Family for Sunflowers

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 $k$.

32 pages. Please visit https://sites.psu.edu/sunflowerconjecture/2022/12/18/index-page/ for additional extensive information