On a Problem Concerning the Weight Functions

Let X be a finite set with n elements. A function f: X − → R such that ∑x ∈ Xf(x) ≥ 0 is called an -weight function. In 1988 Manickam and Singhi conjectured that, if d is a positive integer and f is an -weight function with n ≥ 4 d there exist at least (n − 1 d − 1) subsets Y of X with |Y | = d for which ∑y ∈ Yf(y) ≥ 0. In this paper we study this conjecture and we show that it is true if f is a n -weight function and |{x ∈ X: f(x) ≥ 0}| ≤ d ≤ n 2 .