TY - JOUR

T1 - A note on the Manickam-Miklós-Singhi conjecture

AU - Chowdhury, Ameera

PY - 2014/1

Y1 - 2014/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.ejc.2013.06.010

DO - 10.1016/j.ejc.2013.06.010

M3 - Article

AN - SCOPUS:84882595068

VL - 35

SP - 131

EP - 140

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

ER -