Maximal 3-wise intersecting families (Q6143925)

A family \(\mathcal{F}\) of subsets of \([n]\) is \(k\)-wise intersecting if the intersection of every collection of \(k\) distinct sets in \(\mathcal{F}\) is non-empty. \(\mathcal{F}\) is called maximal \(k\)-wise intersecting if \(\mathcal{F}\) is \(k\)-wise intersecting and no set from \(2^{[n]} \setminus \mathcal{F}\) can be added to \(\mathcal{F}\) keeping the \(k\)-wise intersecting property.\par \textit{P. Erdős} and \textit{D. J. Kleitman} [Discrete Math. 8, 281--294 (1974; Zbl 0281.04002)] ask for the size of the smallest maximal \(k\)-wise intersecting family. This question is answered in the paper under review for \(k=3\).\par The main result of the paper states that if \(n\) is sufficiently large and \(\mathcal{F}\) is a maximal \(3\)-wise intersecting family of minimum size on the ground set \([n]\), then \(\mathcal{F}\) is of the form \(\{A: S \subsetneq A \subseteq[n]\} \cup\left\{B: S^c \subsetneq B \subseteq[n]\right\}\) for some \(S \subseteq[n]\) with \(\left\lfloor\frac{n}{2}\right\rfloor \le \vert S\vert \le \left\lceil\frac{n}{2}\right\rceil\), which the authors call a balanced pair of linked cubes. A key step in the proof is to establish that a maximal 3-wise intersecting family of size at most \((1 + o(1))(2^{\lceil n/2 \rceil} + 2^{\lfloor n/2 \rfloor})\) must be close in structure to a balanced pair of linked cubes.