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    A linear bound on the Manickam--Miklós--Singhi conjecture

    Pokrovskiy, Alexey (2015) A linear bound on the Manickam--Miklós--Singhi conjecture. Journal of Combinatorial Theory, Series A 133 , pp. 280-306. ISSN 0097-3165.

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    Abstract

    Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1 \choose k-1). This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n > 33k^2. In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n > Ck.

    Metadata

    Item Type: Article
    Keyword(s) / Subject(s): Extremal combinatorics, Hypergraphs, Additive combinatorics, Katona's cycle method
    School: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 14:54
    Last Modified: 28 Jun 2020 05:28
    URI: http://eprints.bbk.ac.uk/id/eprint/25899

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