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: | Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School |
| Depositing User: | Alexey Pokrovskiy |
| Date Deposited: | 21 Jan 2019 14:54 |
| Last Modified: | 02 Aug 2023 17:47 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/25899 |
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