A k-permutation family on n vertices is a set-system consisting of the intervals of k permutations of the integers 1 to n. The discrepancy of a set-system is the minimum over all red–blue vertex colourings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any 3-permutation family is at most a constant independent of n. Here we give a simpler proof that Newman and Nikolov’s sequence of 3-permutation families has discrepancy $\Omega (\log \,n)$. We also exhibit a sequence of 6-permutation families with root-mean-squared discrepancy $\Omega (\sqrt {\log \,n} )$; that is, in any red–blue vertex colouring, the square root of the expected squared difference between the number of red and blue vertices in an interval of the system is $\Omega (\sqrt {\log \,n} )$.

中文翻译：

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贝克三排列猜想的简化证明及其在均方根误差上的应用

一种ķ-排列家庭nvertices 是一个集合系统，由ķ整数 1 到的排列n. 集合系统的差异是系统中任何集合中红色和蓝色顶点数量之间的最大差异的所有红色-蓝色顶点着色的最小值。2011 年，Newman 和 Nikolov 推翻了 Beck 的猜想，即任何 3 排列族的差异至多是一个独立于n. 这里我们给出一个更简单的证明，证明 Newman 和 Nikolov 的 3 置换族序列存在差异$\欧米茄 (\log \,n)$. 我们还展示了一系列具有均方根差异的 6 排列族$\Omega (\sqrt {\log \,n} )$; 也就是说，在任何红色-蓝色顶点着色中，系统间隔内红色和蓝色顶点数之间的期望平方差的平方根是$\Omega (\sqrt {\log \,n} )$.