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

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 $1/2$.