Odd hypergraph Mantel theorems

Abstract

A classical result of Sidorenko (1989) shows that the Turán density of every $r$-uniform hypergraph with three edges is bounded from above by $1/2$. For even $r$, this bound is tight, as demonstrated by Mantel's theorem on triangles and Frankl's theorem on expanded triangles. In this note, we prove that for odd $r$, the bound $1/2$ is never attained, thereby answering a question of Keevash and revealing a fundamental difference between hypergraphs of odd and even uniformity. Moreover, our result implies that the expanded triangles form the unique class of three-edge hypergraphs whose Turán density attains $1/2$.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof : Proof of Lemma \ref{['LEMMA:suspension-Turan-density']}
• Lemma 2.2
• Theorem 2.3: KS05
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['LEMMA:describe-T2']}
• Lemma 3.3