Partition-Crossing Hypergraphs

Csilla Bujtás, Z. Tuza

Research output: Contribution to journalArticle

Abstract

For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . ., Xk) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Turán-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.

Original languageEnglish
Pages (from-to)815-828
Number of pages14
JournalActa Cybernetica
Volume23
Issue number3
DOIs
Publication statusPublished - Jan 1 2018

Keywords

  • Crossing set
  • Hypergraph
  • Partition
  • Set system
  • Turán-type problem
  • Upper chromatic number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Computer Science (miscellaneous)
  • Computer Vision and Pattern Recognition
  • Management Science and Operations Research
  • Information Systems and Management
  • Electrical and Electronic Engineering

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