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 − 1 k − 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.