A family F of subsets of {1,…,n} is called k-wise intersecting if any k members of F
Nearby in the stack
have non-empty intersection, and it is called maximal
k
-wise intersecting if no family strictly containing
F
satisfies this condition. We show that for each
k≥2
there is a maximal
k
-wise intersecting family of size
O(2n/(k−1))
. Up to a constant factor, this matches the best known lower bound, and answers an old question of Erdős and Kleitman, recently studied by Hendrey, Lund, Tompkins, and Tran.