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arXiv · 2411.01782

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

Maya Sankar
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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 $1/2$.

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Maya Sankar. 2024-11-04. The Tur\'an Density of 4-Uniform Tight Cycles. https://doi.org/10.19086/aic.2026.4

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