A note on saturation for $k$-wise intersecting families

A family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is called $k$-wise intersecting if any $k$ members of $\mathcal{F}$ have non-empty intersection, and it is called maximal $k$-wise intersecting if no family strictly containing $\mathcal{F}$ satisfies this condition. We show that for each $k\geq 2$ there is a maximal $k$-wise intersecting family of size $O(2^{n/(k-1)})$. Up to a constant factor, this matches the best known lower bound, and answers an old question of Erd\H{o}s and Kleitman, recently studied by Hendrey, Lund, Tompkins, and Tran.Comment: 4 pages; added a new section about the non-existence of certain types of construction