AI 中文总结
解决Han和Zhao关于k - 一致超图中哈密顿\(\ell\) - 圈余度阈值猜想的剩余情况，证明对于特定条件下的\(k\)、\(\ell\)及足够大且可被\(k - \ell\)整除的\(n\)，满足一定条件的超图含哈密顿\(\ell\) - 圈，基于前人框架改进论证。
AI 中文摘要
在本笔记中，我们解决了Han和Zhao关于k - 一致超图中哈密顿\(\ell\) - 圈余度阈值猜想的剩余开放情况。具体而言，我们证明对于整数\(k\geq3\)，\(3k/4\leq\ell<k\)，\(k\not\equiv0\pmod{k - \ell}\)，且对于所有足够大的可被\(k - \ell\)整除的\(n\)，每个满足\(\delta_{k - 1}(H)\geq\frac{n}{(k - \ell)\left\lceil\frac{k}{k - \ell}\right\rceil}\)的\(n\) - 顶点\(k\) - 一致超图\(H\)都包含一个哈密顿\(\ell\) - 圈。我们的证明基于Gan、Han和Xu的框架，并改进他们的论证以在精确阈值处获得所需的路径族。
英文摘要
In this note, we resolve the remaining open case of a conjecture by Han and Zhao concerning the codegree threshold for Hamilton $\ell$-cycles in $k$-uniform hypergraphs. Specifically, we prove that for integers $k\ge 3$, $3k/4\le \ell<k$, with $k\not\equiv 0 \pmod{k-\ell}$, and for all sufficiently large $n$ divisible by $k-\ell$, every $n$-vertex $k$-uniform hypergraph $H$ satisfying \[ δ_{k-1}(H)\ge \frac{n}{(k-\ell)\left\lceil \frac{k}{k-\ell}\right\rceil} \] contains a Hamilton $\ell$-cycle. Our proof builds on the framework of Gan, Han and Xu, and refines their argument to obtain, at the exact threshold, the required family of paths.
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