The Tur\'an Density of 4-Uniform Tight Cycles

TL;DR

This paper characterizes hypergraphs avoiding certain tight cycles using vertex tuple colorings, generalizes previous results, and establishes Turán density for large cycles in specific cases.

Contribution

It provides an exact characterization of hypergraphs avoiding tight cycles modulo r and introduces a strategy to determine Turán densities for large tight-cycle-like hypergraphs.

Findings

Characterization of hypergraphs avoiding tight cycles via colorings.

Extension of bipartite graph avoidance to hypergraphs.

Proof that large tight cycles have Turán density 1/2 under certain conditions.

Abstract

For any uniformity $r$ and residue $k$ modulo $r$, we give an exact characterization of the $r$-uniform hypergraphs that homomorphically avoid tight cycles of length $k$ modulo $r$, in terms of colorings of $(r-1)$-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kam\v cev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form $\mathscr C_k^{(r)}=\{\text{$r$-uniform tight cycles of length $k$ modulo $r$}\}$. We also outline a general strategy to prove that, if $\mathscr C$ is a family of tight-cycle-like hypergraphs (including but not limited to the families $\mathscr C_k^{(r)}$) for which the above characterization applies, then all sufficiently long $C\in \mathscr C$ will have the same Tur\'an density. We demonstrate an application of this framework, proving that there exists an integer $L_0$ such that for every $L>L_0$ not divisible by 4, the tight cycle $C^{(4)}_L$ has Tur\'an density $1/2$.