An improved bound for the Manickam–Miklós–Singhi conjecture

AbstractWe show that for n>k2(4elogk)k, every set {x1,⋯,xn} of n real numbers with ∑i=1nxi≥0 has at least (n−1k−1)k-element subsets of a non-negative sum. This is a substantial improvement on the best previously known bound of n>(k−1)(kk+k2)+k, proved by Manickam and Miklós [9] in 1987

Lattices of partial sums

In this paper we introduce and study a class of partially ordered sets that can be interpreted as partial sums of indeterminate real numbers. An important example of these partially ordered sets, is the classical Young lattice Y of the integer partitions. In this context, the sum function associated to a specific assignment of real values to the indeterminate variables becomes a valuation on a distributive lattice