发表机构
Royal Holloway University of London; Nankai University（伦敦大学皇家霍洛威学院; 南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了满足半度条件但无法嵌入高度对称锦标赛的近生成平衡反定向树，给出了Stein猜想在均匀次线性度解释下的反例，并确定了最优目标阶及最小宿主阶。
AI 中文摘要
我们构造了近生成平衡反定向树，这些树在满足猜想中的半度条件的情况下仍无法嵌入高度对称的锦标赛。更精确地说，对于每个足够大的奇数整数$n$，我们构造了一个$n$个顶点上的正则顶点传递锦标赛，以及一个$n-1$个顶点上的平衡反定向毛毛虫，其最大度至多为$(1+o(1))n/\log_2 n$，且该毛毛虫不包含在该锦标赛中。宿主在半度和伪半度设置下均满足严格的$k/2$阈值，其中$k$是目标树的弧数。因此，这为Stein综述（2024）中猜想6.8和7.6的均匀次线性度解释提供了反例。目标的阶$n-1$在严格半度假设下是最优的。我们还给出了一个六顶点平衡反定向双星，它缺失于七顶点Paley锦标赛中，并证明了七是该固定双星在伪半度反例中的最小宿主阶。
英文摘要
We construct near-spanning balanced antidirected trees that fail to embed in highly symmetric tournaments despite the conjectured semidegree condition being satisfied. More precisely, for every sufficiently large odd integer $n$, we exhibit a regular vertex-transitive tournament on $n$ vertices and a balanced antidirected caterpillar on $n-1$ vertices with maximum degree at most $(1+o(1))n/\log_2 n$ that is not contained in the tournament. The host satisfies the strict $k/2$ threshold in both the semidegree and pseudo-semidegree settings, where $k$ is the number of arcs of the target tree. Consequently, this gives counterexamples to the uniform sublinear-degree interpretations of Conjectures~6.8 and~7.6 in Stein's survey (2024). The order $n-1$ of the target is best possible under the strict semidegree hypothesis. We also give a six-vertex balanced antidirected double-star missing from the seven-vertex Paley tournament, and show that seven is the smallest host order for a pseudo-semidegree counterexample with this fixed double-star.
Comments6 pages
