The structure of maximal non-trivial d-wise intersecting uniform families with large sizes

Abstract

For a positive integer d 2, a family F ⊆ [n] k is said to be d-wise intersecting ifWe provide a refinement of O'Neill and Verstraëte's Theorem about the structure of the largest and the second largest maximal non-trivial d-wise intersecting k-uniform families. We also determine the structure of the third largest and the fourth largest maximal non-trivial d-wise intersecting k-uniform families for any k > d + 1 4, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any k 5, in the asymptotic sense. Our proofs are applications of the ∆-system method.