主要锦标赛家族
Dominant tournament families
摘要：
对于具有$h$顶点的锦标赛$H$，其典型密度是$h!2^{-\binom{h}{2}}/aut(H)$，即，这是随机锦标赛中$H$的期望密度。一个$h$ -顶点竞赛图族${\mathcal F}$是{ \em显性}，如果对于所有足够大的$n$，存在一个$n$ -顶点竞赛图$G$，使得$G$中${\mathcal F}$的每个元素的密度比其典型密度大一个常数因子。对于小$h$来说，表征所有优势家族已经是一项挑战。在这里，我们描述了几个大的优势家庭，每$h$。特别是，我们证明了以下所有$h$足够大：（i）对于所有的比赛$H^*$至少有$5\log h$顶点，家庭的所有$h$ -顶点比赛，包含$H^*$作为一个子图是占主导地位的。(ii)所有最小反馈弧集大小至多为$\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$的$h$ -顶点竞赛图族是占优的。对于小的$h$，我们构造了$6$的支配族（即$50\%$的）$5$顶点上的竞赛图和$h=6,7,8,9$的尺寸大于$40\%$的支配族。对于所有的$h$，我们提供了一个显式的结构的一个占主导地位的家庭，这是为了获得一个绝对常数分数的比赛$h$顶点。一些额外的有趣的开放问题。
英文摘要：
For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this is the expected density of $H$ in a random tournament. A family ${\mathcal F}$ of $h$-vertex tournaments is {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex tournament $G$ such that the density of each element of ${\mathcal F}$ in $G$ is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small $h$. Here we characterize several large dominant families for every $h$. In particular, we prove the following for all $h$ sufficiently large: (i) For all tournaments $H^*$ with at least $5\log h$ vertices, the family of all $h$-vertex tournaments that contain $H^*$ as a subgraph is dominant. (ii) The family of all $h$-vertex tournaments whose minimum feedback arc set size is at most $\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$ is dominant. For small $h$, we construct a dominant family of $6$ (i.e. $50\%$ of the) tournaments on $5$ vertices and dominant families of size larger than $40\%$ for $h=6,7,8,9$. For all $h$, we provide an explicit construction of a dominant family which is conjectured to obtain an absolute constant fraction of the tournaments on $h$ vertices. Some additional intriguing open problems are presented.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
