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

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A 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.