AI 中文总结
本文在最小顶点度和最小共度数条件下，确定均匀稠密3 - 图中紧密哈密顿圈的尖锐对角阈值，证明\(d > 1/3\)且\(\alpha > f(d)\)时相关密度和度条件迫使存在紧密哈密顿圈，还给出最小共度数条件下阈值，反驳了一些猜想。
AI 中文摘要
一个在n个顶点上的3 - 均匀超图（或3 - 图）H，如果对于所有\(X,Y,Z\subseteq V(H)\)，有\(e_H(X,Y,Z)\geq d|X||Y||Z| - \mu n^3\)，则称其为\((n,d,\mu)\) - 稠密的。这是3 - 图准随机性最弱的标准概念之一，也称为线性准随机性。在本文中，我们在最小顶点度\(\delta_1(H)\)和最小共度数\(\delta_2(H)\)的条件下，确定了\((n,d,\mu)\) - 稠密3 - 图H中紧密哈密顿圈的尖锐对角阈值。我们实际证明了一个一般结果：定义\(f(d):=\frac{1 - \sqrt{(4d - 1)/3}}{2}\)。我们证明，当\(d > 1/3\)且\(\alpha > f(d)\)时，\((n,d,\mu)\) - 密度以及\(\delta_1(H)\geq\alpha\binom{n - 1}{2}\)迫使存在一个紧密哈密顿圈。特别地，\(f(1/3)=1/3\)，这回答了Araújo、Piga和Schacht的问题8.3(i)并证实了Han、Shu和Wang的猜想8.1。对于最小共度数条件，尖锐对角阈值是\((\kappa,\kappa)\)，其中\(\kappa\)是\(\kappa=(1 - \kappa)^3\)的唯一实解。由于\(\kappa\approx0.3177 > 1/4\)，这给出了对Araújo、Piga和Schacht问题8.3(ii)的否定答案，并反驳了Han、Shu和Wang的猜想8.2。两个证明使用了共同的哈密顿框架约简，但两个度条件导致了\((n,d,\mu)\) - 稠密3 - 图的不同主导分量引理，这些引理具有独立的研究价值，且其证明不依赖于吸收方法。
英文摘要
A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \emph{$(n,d,μ)$-dense} if $e_H(X,Y,Z)\ge d|X||Y||Z|-μn^3$ for all $X,Y,Z\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness. In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,μ)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $δ_1(H)$ and the minimum codegree $δ_2(H)$. We actually prove a general result: define \[ f(d):=\frac{1-\sqrt{(4d-1)/3}}2. \] We prove that $(n,d,μ)$-density together with $δ_1(H)\geα\binom{n-1}{2}$ forces a tight Hamilton cycle whenever $d > 1/3$ and $α>f(d)$. In particular, $f(1/3)=1/3$, which answers Problem~8.3(i) of Araújo, Piga and Schacht and confirms Conjecture~8.1 of Han, Shu and Wang. For the minimum codegree condition, the sharp diagonal threshold is $(κ,κ)$, where $κ$ is the unique real solution of $κ=(1-κ)^3$. Since $κ\approx0.3177>1/4$, this gives a negative answer to Problem~8.3(ii) of Araújo, Piga and Schacht and disproves Conjecture~8.2 of Han, Shu and Wang. The two proofs use a common Hamilton-framework reduction, but the two degree conditions lead to distinct dominant-component lemmas for $(n, d, μ)$-dense $3$-graphs, which are of independent interest and whose proofs do not rely on the absorption method.
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