Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

A triangle is a family of three sets A,B,C such that AB, BC, CA are each nonempty, and \( A \cap B \cap C = \emptyset \). Let \( {\user1{\mathcal{A}}} \) be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have \( {\left| {\user1{\mathcal{A}}} \right|} \leqslant {\left( {\begin{array}{*{20}c} {{n - 1}} \\ {{r - 1}} \\ \end{array} } \right)}. \) This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. We also show that equality holds if and only if \( {\user1{\mathcal{A}}} \) consists of all r-element subsets containing a fixed element.

Analogous results are obtained for nonuniform families.

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Correspondence to Dhruv Mubayi.

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Mubayi, D., Verstraëte, J. Proof Of A Conjecture Of Erdős On Triangles In Set-Systems. Combinatorica 25, 599–614 (2005).

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Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05