Abstract
Suppose that we have a set S of n real numbers which have nonnegative sum. How few subsets of S of order k can have nonnegative sum? Manickam, Miklós, and Singhi conjectured that for n≥4k the answer is (n−1k−1). This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n≥33k2. In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n≥Ck. This establishes the conjecture in a range which is a constant factor away from the conjectured bound.
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