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

A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture

“…Moreover, as pointed out in [22], this conjecture settles some cases of another conjecture on multiplicative functions by Alladi, Erdös and Vaaler, [2]. Partial results related to the Manickam-Miklös-Singhi conjecture have been obtained also in [5], [6], [11], [12], [13]. Now, if 1 ≤ r ≤ n, we set: (1) γ(n, r) = min{α(f ) : f ∈ W n (R), f + = r},…”

“…The numbers γ(n, d, r) have been introduced in [11] and they also have been studied in [12], in order to solve the Manickam-Miklös-Singhi conjecture, because it is obviuos that:…”

The Manickam–Miklós–Singhi conjectures for sets and vector spaces

“…Although Conjecture 1.1 and the Erdős-Ko-Rado theorem share the same bound and extremal example, there is no obvious way to translate one question into the other. Conjecture 1.1 has attracted a lot of attention due to its connections with the Erdős-Ko-Rado theorem [1,2,4,6,7,8,9,10,11,12,15,18,19,22,23,24,25,26,27], but still remains open. For more than two decades, Conjecture 1.1 was known to hold only when k|n [24] or when n is at least an exponential function of k [6,8,23,27].…”

Sand Piles Models of Signed Partitions with Piles

“…In [22] the authors have introduced and studied a poset ( ( , , related to some extremal combinatorial sums problems (see also [23][24][25][26] for studies on these problems). Such a poset can be seen as a lattice of particular integer partitions with all distinct summands which can be positive or negative, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than −( − .…”