Extensions of a Family for Sunflowers

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 $$Ext(\mathcal{F},l)=\{V:V\in \left(\genfrac{}{}{0px}{}{X}{l}\right),\mathrm{\exists }U\in \mathcal{F}U\subset V\}$$ 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 $\mathrm{\Gamma }(b)$-condition, that is, $$\mid \mathcal{F}[S]\mid <b^{-\mathrm{\mid }S\mathrm{\mid }}\mathrm{\mid }\mathcal{F}\mathrm{\mid }\text{for every nonempty set }S,\text{where}\mathcal{F}[S]:=\{U:U\in \mathcal{F},S\subset U\},$$ for $b=m^{\frac{1}{2}+\epsilon }$ with an $m$ sufficiently larger than a given constant $1\mathrm{/}\epsilon$. It is stronger than the statement that $\mathcal{F}$ includes a 3-sunflower if $\mathrm{\mid }\mathcal{F}\mathrm{\mid }>b^m$, 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 $\mathrm{\Gamma }\left(8^{\sqrt{{\log }_{2}m}}\sqrt{m}k{\log }_{2}k\right)$-condition with an $m$ sufficiently larger than $k$.