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

For a positive integer d⩾2, family F⊆([n]k) is said to be d-wise intersecting if |F1∩F2∩…∩Fd|⩾1 for all F1,F2,…,Fd∈F. A called maximal F∪{A} not any A∈([n]k)∖F. We provide refinement of O'Neill and Verstraëte's Theorem about the structure largest second non-trivial k-uniform families. also determine third fourth families k>d+1⩾4, fifth sixth 3-wise k⩾5, in asymptotic sense. Our proofs are applications Δ-system method.