发表机构
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对 Bollobás 和 Scott 提出的两锦标赛偏差是否恒为 $\u03a9(n^{3/2})$ 的问题，证明了存在绝对常数 $c>0$，使任意两阶数为 $n$ 的锦标赛偏差至少为 $cn^{3/2}$，解答了该问题。
AI 中文摘要
给定两个阶数为 $n$ 的锦标赛，Bollobás 和 Scott 在将它们在公共顶点集上重新标记后，根据它们的重叠与随机平均值的极值偏差定义了它们的偏差，他们询问由此得到的偏差是否恒为 $\u03a9(n^{3/2})$。我们通过证明存在绝对常数 $c>0$，使得每对阶数为 $n$ 的锦标赛 $T,U$ 的偏差至少为 $cn^{3/2}$，从而回答了该问题。
英文摘要
Given two tournaments of order $n$, Bollobás and Scott defined their discrepancy as the largest deviation of their overlap from its random average under relabelling, and they asked whether the resulting discrepancy is always $Ω(n^{3/2})$. We answer this question by proving that there is an absolute constant $c>0$ such that every pair of tournaments $T,U$ of order $n$ has discrepancy at least $cn^{3/2}$. More generally, if $D$ and $H$ are oriented graphs of order \( n \) with \( e(D) = p \binom{n}{2} \) and \( e(H) = q \binom{n}{2} \) satisfying $16/n \leq p, q \leq 1 - 16/n$, then there is an absolute constant \( c > 0 \) such that their discrepancy is at least $c(p(1-p)q(1-q))^{3}n^{3/2}$. We also show that these two-graph estimates extend to intersections of any fixed number of graphs, tournaments, and oriented graphs.
