Minimal paths and cycles in set systems

@article{Mubayi2007MinimalPA,
  title={Minimal paths and cycles in set systems},
  author={Dhruv Mubayi and Jacques Verstra{\"e}te},
  journal={Eur. J. Comb.},
  year={2007},
  volume={28},
  pages={1681-1693}
}
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(n) ≤ fr(n, k) ≤ 3` ( n− 1 r − 1 ) + O(n), 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… CONTINUE READING

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