On set systems with a threshold property

Zoltán Füredi, Robert H. Sloan, Ken Takata, György Turán

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Abstract

For n, k and t such that 1 < t < k < n, a set F of subsets of [n] has the (k, t)-threshold property if every k-subset of [n] contains at least t sets from F and every (k - 1)-subset of [n] contains less than t sets from F. The minimal number of sets in a set system with this property is denoted by m (n, k, t). In this paper we determine m (n, 4, 3) exactly for n sufficiently large, and we show that m (n, k, 2) is asymptotically equal to the generalized Turán number Tk - 1 (n, k, 2).

Original languageEnglish
Pages (from-to)3097-3111
Number of pages15
JournalDiscrete Mathematics
Volume306
Issue number23 SPEC. ISS.
DOIs
Publication statusPublished - Dec 6 2006

Keywords

  • Extremal problem
  • Packing
  • Set system

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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    Füredi, Z., Sloan, R. H., Takata, K., & Turán, G. (2006). On set systems with a threshold property. Discrete Mathematics, 306(23 SPEC. ISS.), 3097-3111. https://doi.org/10.1016/j.disc.2006.06.001