A remark on the problem of nonnegative k-subset sums

Abstract

Given a set of n real numbers with a nonnegative sum, consider the family of all its k-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in [1]. The latter, in a special case, is nothing else but the problem of determining a minimal number c n (k) such that any k-uniform hypergraph on n vertices having c n (k) + 1 edges has a perfect fractional matching. We show that results obtained in [1] can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution.