New results related to a conjecture of Manickam and Singhi

Abstract

In 1998 Manickam and Singhi conjectured that for every positive integer $d$ and every $n\ge 4d$, every set of $n$ real numbers whose sum is nonnegative contains at least $\left(\genfrac{}{}{0ex}{}{n-1}{d-1}\right)$ subsets of size $d$ whose sums are nonnegative. In this paper we establish new results related to this conjecture. We also prove that the conjecture of Manickam and Singhi does not hold for $n=2d+2$.

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