On the minimum size of maximal k-wise intersecting families · arXivDeskAbstract
A family F of subsets of [n]:={1,2,…,n} is called maximal k-wise intersecting if every collection of at most k
members of
has a non-empty intersection, and adding any other set to
breaks this property. An old question by Erdős and Kleitman from 1974 asks for the minimum size of a maximal
-wise intersecting family. The case
is known for all sufficiently large
, but the problem remains open for all
. The previous best-known upper bound is by Janzer, which has a leading term
(k−1)2k−32n/(k−1) for sufficiently large
divisible by
. In this note, we improve this bound to
(4k−10)2n/(k−1) , which reduces the dependence on
in the leading coefficient from exponential to linear and is within a factor of
of the known lower bound.