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arXiv · 2606.22828

Tur\'an numbers of $4$-uniform tight even cycles minus one edge

Wanfang Chen·Jianfeng Hou·Xizhi Liu·Yixiao Zhang·Hongbin Zhao
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Abstract

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Tur\'{a}n number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash--Sudakov on the Tur\'an density and the Tur\'{a}n number of the $4$-uniform expanded triangle. We also show that the Tur\'{a}n density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Tur\'{a}n density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

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Wanfang Chen, Jianfeng Hou, Xizhi Liu, Yixiao Zhang, Hongbin Zhao. 2026-06-22. Tur\'an numbers of $4$-uniform tight even cycles minus one edge. https://arxiv.org/abs/2606.22828

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