New results related to a conjecture of Manickam and Singhi

In 1988 Manickam and Singhi conjectured that for every positive integer dd and every n≥4dn≥4d, every set of nn real numbers whose sum is non-negative contains at least (n−1d−1) subsets of size dd whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture.