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A simplified disproof of Beck’s three permutations conjecture and an application to root-mean-squared discrepancy

Part of: Extremal combinatorics Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  26 October 2020

Cole Franks
Cole Franks*
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ08854, USA Email: franks@mit.edu
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Abstract

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 $\Omega (\sqrt {\log \,n} )$.

MSC classification

Primary: 68R01: General
Secondary: 05D99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 3 , May 2021 , pp. 398 - 411
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Supported in part by Simons Foundation award 332622.

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