Disjoint pairs in set systems with restricted intersection

Abstract

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In particular, we show that for any pair of set systems (A, B) which avoid a cross-intersection of size t, the number of disjoint pairs (A, B) with A ∈ A and B ∈ B is at most t−1 k=0 n k 2 n−k . This implies an asymptotically best possible upper bound on the number of disjoint pairs in a single t-avoiding family F ⊂ P[n]. We also study this problem when A, B ⊂ [n] (r) are both r-uniform, and show that it is closely related to the problem of determining the maximum of the product |A||B| when A and B avoid a cross-intersection of size t, and n ≥ n 0 (r, t).

On an inverse problem of the Erdős-Ko-Rado type theorems

“…Starting from this, the stability and supersaturation for extremal families are then well worth studying. In recent years, there have been a lot of works concerning this kind of inverse problems, for examples, see [3,4,[8][9][10]18,26,29,34].…”

Inverse problems of the Erdős-Ko-Rado type theorems for families of vector spaces and permutations

“…This gives rise to further studies of the stability and supersaturation for extremal families. In recent years, there have been a lot of works concerning this kind of inverse problems, for examples, see [1,2,8,9,11,17,19,22,29].…”