An Extension of Mantel’s Theorem to k-Graphs

Zoltán Füredi, András Gyárfás

Research output: Contribution to journalComment/debate

Abstract

According to Mantel’s theorem, a triangle-free graph on n points has at most n2/4 edges. A linear k-graph is a set of points together with some k-element subsets, called edges, such that any two edges intersect in at most one point. The k-graph Fk, called a fan, consists of k edges that pairwise intersect in exactly one point v, plus one more edge intersecting each of these edges in a point different from v. We extend Mantel’s theorem as follows: fan-free linear k-graphs on n points have at most n2/k2 edges. This extension nicely illustrates the difficulties of hypergraph Turán problems. The determination of the case of equality leads to transversal designs on n points with k groups—for k = 3 these are equivalent to Latin squares. However, in contrast to the graph case, new structures and open problems emerge when n is not divisible by k.

Original languageEnglish
Pages (from-to)263-268
Number of pages6
JournalAmerican Mathematical Monthly
Volume127
Issue number3
DOIs
Publication statusPublished - Mar 15 2020

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Keywords

  • 05C70
  • MSC: Primary 05D05
  • Secondary 05B15

ASJC Scopus subject areas

  • Mathematics(all)

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