On hierarchically closed fractional intersecting families (Q6117257)

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