On hierarchically closed fractional intersecting families

TL;DR

This paper investigates fractional intersecting families with hierarchical closure properties, establishing linear size bounds for certain parameters and providing exact bounds and classifications for the case when the fraction is 1/2.

Contribution

It introduces the concept of fractional r-closed L-intersecting families and proves size bounds, including tight bounds and uniqueness results for specific cases.

Findings

For r ≥ 3 and L = {θ}, such families have size at most linear in n.

When θ = 1/2, the maximum size is exactly ⌊3n/2⌋ - 2.

Maximal 1/2-intersecting families are unique up to isomorphism.

Abstract

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $\theta \in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ \theta \lvert A_1 \rvert, \dotsc, \theta \lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{\theta\}$ any fractional $r$-closed $\theta$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $\theta = 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.