Minimal paths and cycles in set systems

A minimal k -cycle is a family of sets A 0 , … , A k − 1 for which A i ∩ A j ≠ 0̸ if and only if i = j or i and j are consecutive modulo k . Let f r ( 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 , ℓ n − 1 r − 1 + O ( n r − 2 ) ≤ f r ( n , k ) ≤ 3 ℓ n − 1 r − 1 + O ( n r − 2 ) , where ℓ = ⌊ ( k − 1 ) / 2 ⌋ . We also prove that f r ( n , 4 ) = ( 1 + o ( 1 ) ) n − 1 r − 1 as 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.