A simplified disproof of Beck's three permutations conjecture and an application to root-mean-squared discrepancy
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A k-permutation family on n vertices is a set system consisting of the intervals of k permutations of the integers 1 through n. The discrepancy of a set system is the minimum over all red-blue vertex colorings 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 Ω( n). We also exhibit a sequence of 6-permutation families with root-mean-squared discrepancy Ω(√( n)); that is, in any red-blue vertex coloring, the square root of the expected difference between the number of red and blue vertices in an interval of the system is Ω(√( n)).
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