Non-trivial 3-wise intersecting uniform families

A family of k-element subsets of an n-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every epsilon > 0 there exists n(0) such that if n > n(0) and 2/5 +epsilon < k/n < 12 - epsilon then the maximum size is 4((n-4)(k-3)) + ((n-4)(k-4)). (c) 2023 Elsevier B.V. All rights reserved.