AI 中文总结
该研究针对锦标赛中有向树的有限界问题，改进中位数排序方法，结合多叶子与少叶子情况的界，将通用界系数从21/8降至18/7。
AI 中文摘要
Sumner的通用锦标赛猜想断言，每个有2n-2个顶点的锦标赛都包含每个有n个顶点的有向树。设f(n)为最小整数N，使得每个有N个顶点的锦标赛都包含每个有n个顶点的有向树。Havet和Thomassé证明f(n)≤⌈(7n-5)/2⌉，El Sahili将其改进为f(n)≤3n-3，Dross和Havet随后得到f(n)≤⌈21n/8 - 47/16⌉。本文改进了他们的中位数排序方法，更准确地说，每个有n个顶点、k个叶子的非双支树（non-bi-arborescence）是(4n-2k-4)-不可避免的，这严格改进了他们针对多叶子情况的估计；双支树（bi-arborescences）满足更强的界2n-2。将此改进与他们针对少叶子情况的界结合，得到对于所有n≥2，f(n)≤⌈(18n-23)/7⌉，因此之前适用于所有n的最优通用界中的系数从21/8降至18/7。
英文摘要
Sumner's universal tournament conjecture asserts that every tournament on $2n-2$ vertices contains every oriented tree on $n$ vertices. Let $f(n)$ be the least integer $N$ such that every tournament on $N$ vertices contains every oriented tree on $n$ vertices. Havet and Thomassé proved that $f(n)\le \lceil(7n-5)/2\rceil$, El Sahili improved this to $f(n)\le3n-3$, and Dross and Havet subsequently obtained $f(n)\le\lceil21n/8-47/16\rceil$. We refine their median-order method. More precisely, every non-bi-arborescence on $n$ vertices with $k$ leaves is $(4n-2k-4)$-unavoidable, which strictly improves their many-leaf estimate; bi-arborescences satisfy the stronger bound $2n-2$. Combining this refinement with their few-leaf bound gives $f(n)\le\lceil(18n-23)/7\rceil$ for every $n\ge2$. Thus the coefficient in the previously best general bound valid uniformly for all $n$ is reduced from $21/8$ to $18/7$.
