Minimal paths and cycles in set systems

Authors:
Dhruv Mubayi;Jacques Verstraëte
Affiliations:
Department of Mathematical Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Venue:
European Journal of Combinatorics
Year:
2007

Citing 5
Cited 4

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Abstract

A minimal k-cycle is a family of sets A"0,...,A"k"-"1 for which A"i@?A"j0@? 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, @?n-1r-1+O(n^r^-^2)@?f"r(n,k)@?3@?n-1r-1+O(n^r^-^2), where @?=@?(k-1)/2@?. We also prove that f"r(n,4)=(1+o(1))n-1r-1 as n-~. This supports a conjecture of Z. Furedi [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.