Minimal paths and cycles in set systems

A minimal k-cycle is a family of sets A0, . . . , Ak−1 for which Ai ∩ Aj = ∅ 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 = (k − 1)/2 . We also prove that fr(n, 4) = (1 + o(1)) n−1 r−1 as n → ∞. This supports a conjecture of F¨uredi [9] on families in which no two pairs of disjoint sets have the same union.

• Type: Journal
• Year: 2007
• Where: EJC
• Authors: Dhruv Mubayi, Jacques Verstraëte
• Keywords: Consecutive Modulo, Disjoint Sets, EJC 2007, Information Technology, Minimal K-cycle