Minimal paths and cycles in set systems

A minimal k-cycle is a family of sets A0, . . . , Ak−1 for which Ai ∩Aj 6= ∅ if and only if i = j or i and j are consecutive modulo k. Let fr(n, k) be the maximum size of a family of r-sets of an n element set containing no minimal k-cycle. Our results imply that for fixed r, k ≥ 3, ` ( n− 1 r − 1 ) +O(nr−2) ≤ fr(n, k) ≤ 3` ( n− 1 r − 1 ) +O(nr−2), where ` = b(k− 1)/2c. We also prove that fr(n, 4) = (1 + o(1)) ( n−1 r−1 ) as n→∞. This supports a conjecture of Füredi [9] on families in which no two pairs of disjoint sets have the same union.