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 $k$.
Categories: math.CO