  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.
