A note on the Manickam-Miklós-Singhi conjecture

Ameera Chowdhury

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

For k∈Z+, let f (k) be the minimum integer N such that for all n ≥ N, every set of n real numbers with nonnegative sum has at least (n-1k-1)k-element subsets whose sum is also nonnegative. In 1988, Manickam, Miklós, and Singhi proved that f (k) exists and conjectured that f (k) ≤ 4. k. In this note, we prove f (3) = 11, f (4) ≤ 24, and f (5) ≤ 40, which improves previous upper bounds in these cases. Moreover, we show how our method could potentially yield a quadratic upper bound on f (k) We end by discussing how our methods apply to a vector space analogue of the Manickam-Miklós-Singhi conjecture.

Original languageEnglish (US)
Pages (from-to)131-140
Number of pages10
JournalEuropean Journal of Combinatorics
Volume35
DOIs
StatePublished - Jan 2014

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