Extensions of a Family for Sunflowers
Junichiro Fukuyama

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 in relation to sunflowers. The previous study found some intrinsic properties of the -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 of -cardinality sets. Subsequently, it lead to the proof that such an includes three mutually disjoint sets if it satisfies the -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 with an sufficiently larger than a given constant . It is stronger than the statement that includes a 3-sunflower if $|\mathcal{F}| >…
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Taxonomy
TopicsLimits and Structures in Graph Theory
