锦标赛中不可避免爪的改进界
Improved Bounds for Unavoidable Claws in Tournaments
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中文总结 AI 辅助
该研究改进锦标赛不可避免爪的常数c_claw的上下界，分离参数θ和σ构建统一框架，证明σ≤θ及相关范围，为该领域提供更优界值与新分析框架。
中文摘要 AI 辅助
设u(n)为使得每个含至多d个分支的n顶点爪都出现在每个n顶点锦标赛中的最大整数d，令c_claw=limsupₙ→∞u(n)/n。1998年，Lu、Wang和Wong证明19/50≤c_claw≤11/23，该结果长期为最优界。我们将其改进为2/5≤c_claw≤10/21。还分离出参数θ和σ，将上下界论证置于同一框架：证明σ≤θ且1/21≤σ≤θ≤1/5，两个界为端点经映射α↦1/2−α/2的像，若σ=θ则lim u(n)/n必存在。
英文摘要
Let $u(n)$ be the largest integer $d$ such that every $n$-vertex claw with at most $d$ branches occurs in every tournament on $n$ vertices, and let $c_{\mathrm{claw}}=\limsup_{n\to\infty}u(n)/n$. In 1998, Lu, Wang and Wong proved that $19/50\le c_{\mathrm{claw}}\le11/23$, and these have remained the best bounds known. We improve them to $2/5\le c_{\mathrm{claw}}\le10/21$. We also isolate two parameters $θ$ and $σ$ which place the lower- and upper-bound arguments in a common framework: we show $σ\leθ$ and $1/21\leσ\leθ\le1/5$, our two bounds being the images of the endpoints under $α\mapsto\frac12-\fracα{2}$, and $σ=θ$ would force $\lim u(n)/n$ to exist.
