On hierarchically closed fractional intersecting families

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 $\mid A_1\cap \cdots \cap A_t\mid \in \{\theta \mid A_1\mid ,\dots ,\theta \mid A_t\mid \}$. 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 $\theta =1\mathrm{/}2$ we have a tight upper bound of $\lfloor \frac{3n}{2}\rfloor -2$ and that a maximal $r$-closed $(1\mathrm{/}2)$-intersecting family is determined uniquely up to isomorphism.