On hierarchically closed fractional intersecting families

Abstract

For a set $L$ of positive proper fractions and a positive integer $r\ge 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F}\subset \mathcal{P}([n])$ with the property that for any $2\le t\le r$ and $A_1,\dots ,A_t\in \mathcal{F}$ there exists $\theta \in L$ such that $|A_1\cap \cdots \cap A_t|\in \{\theta |A_1|,\dots ,\theta |A_t|\}$. In this paper we show that for $r\ge 3$ 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.

20 pages, 0 figures. Included addendum

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