Minimal paths and cycles in set systems

A minimal kk-cycle is a family of sets A0,…,Ak−1A0,…,Ak−1 for which Ai∩Aj≠0̸Ai∩Aj≠0̸ if and only if i=ji=j or ii and jj are consecutive modulo kk. Let fr(n,k)fr(n,k) be the maximum size of a family of rr-sets of an nn element set containing no minimal kk-cycle. Our results imply that for fixed r,k≥3r,k≥3, ℓn−1r−1+O(nr−2)≤fr(n,k)≤3ℓn−1r−1+O(nr−2), where ℓ=⌊(k−1)/2⌋ℓ=⌊(k−1)/2⌋. We also prove that fr(n,4)=(1+o(1))n−1r−1 as n→∞n→∞. This supports a conjecture of Z. Füredi [Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (2–3) (1984) 161–168] on families in which no two pairs of disjoint sets have the same union.