AI 中文总结
本文针对均匀超图构造有色偏紧致哈密顿环的层状障碍，通过 $k=17$、$a=8$ 的反例及渐近分析，揭示其内部成员与边界构造的行为差异。
AI 中文摘要
我们在均匀超图中构造了一族针对有色偏紧致哈密顿环的层状障碍。对每个 $k\ge 3$ 和每个 $a\in\{0,\ldots,k-1\}$，我们给出一个红蓝着色的 $k$ 均匀超图，该超图包含一个紧致哈密顿环，而构造中的每个紧致哈密顿环都是完全颜色平衡的。Behague、Clemen、Hyde 和 Morrison 所猜想的更高均匀度阈值对应的构造，是该族的边界情况 $a=0$。我们证明，$a$ 的内部取值可产生更稠密的障碍。特别地，对于 $k=17$ 且 $a=8$，我们构造的渐近相对最小顶点度为 $\frac{5761}{8192}\approx 0.703247$，超过了猜想值 $d_{17}\approx 0.699277$，这构成了对 Behague、Clemen、Hyde 和 Morrison 文献中猜想6.1提出的更高均匀度阈值的反例。此外，通过在 $k\to\infty$ 时适当选择层，该族包含渐近相对最小顶点度为 $1-O\bigl(k^{-1/2}\bigr)$ 的障碍。因此，在大均匀度下，层状障碍族的内部成员表现出与此前考虑的边界构造截然不同的行为。
英文摘要
We construct a family of layer barriers for colour-biased tight Hamilton cycles in uniform hypergraphs. For every $k\ge 3$ and every $a\in\{0,\ldots,k-1\}$, we give a red--blue coloured $k$-graph that contains a tight Hamilton cycle, while every tight Hamilton cycle in the construction is perfectly colour-balanced. The construction underlying the higher-uniformity threshold conjectured by Behague, Clemen, Hyde and Morrison corresponds to the boundary case $a=0$ of this family. We show that interior choices of $a$ can yield strictly denser barriers. In particular, for $k=17$ and $a=8$, the asymptotic relative minimum vertex degree of our construction is \[ \frac{5761}{8192}\approx 0.703247, \] which exceeds the conjectured value $d_{17}\approx 0.699277$. This provides a counterexample to the proposed higher-uniformity threshold in Conjecture~6.1 of Behague, Clemen, Hyde and Morrison. Moreover, by choosing the layer appropriately as $k\to\infty$, the family contains barriers whose asymptotic relative minimum vertex degree is \[ 1-O\bigl(k^{-1/2}\bigr). \] Thus the interior members of the layer-barrier family exhibit substantially different behaviour from the previously considered boundary construction in large uniformity.
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