be n real numbers with non-negative sum. We show that if n ≥ 12 there exist at least n−1 2 subsets of {a 1 , . . . , a n } with three elements which have non-negative sum.

No abstract

No abstract

The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Miklös and Singhi. We study the fundamentals properties of such lattices and of a particular class of boolean functions defined on them.

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 $\binom {n-1}{d-1}$ 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 $n=2d+2$