The Hypergraph Tur\'{a}n Densities of Tight Cycles Minus an Edge

TL;DR

This paper determines the maximum number of edges in 3-uniform hypergraphs avoiding a specific tight cycle minus an edge, providing exact extremal numbers and characterizing extremal graphs for certain cycle lengths.

Contribution

It establishes the exact extremal number for tight cycles minus an edge in 3-graphs for infinitely many cycle lengths and characterizes extremal configurations.

Findings

Exact extremal number for certain cycle lengths.

Extremal graphs characterized up to O(n) edge edits.

Asymptotic density approaches 1/4 of all triples.

Abstract

A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$, the extremal number is $ ex\left(C_\ell^-,n\right)=\tfrac1{24}n^3+O(n\ln n)=\left(\tfrac14+o(1)\right){n\choose 3}. $ We determine the extremal graph up to $O(n)$ edge edits.