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.

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dhruv Mubayi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mubayi, D., Verstraëte, J. Proof Of A Conjecture Of Erdős On Triangles In Set-Systems. Combinatorica 25, 599–614 (2005). https://doi.org/10.1007/s00493-005-0036-0

Download citation

Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05