A note on the Manickam-Miklós-Singhi conjecture

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)≤4k. 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.