Abstract:More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vec… Show more
“…He reduced the conjecture to finding a k-uniform hypergraph on n vertices satisfying the MMS property (similar techniques were also employed earlier in [8]). The second conjecture, Conjecture 1.3, was very recently proved by Chowdhury, Sarkis, and Shahriari [3] simultaneously with our work. They also proved a quadratic bound n ≥ 8k 2 for sets.…”
An r-unform n-vertex hypergraph H is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n > 10r 3 , every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n > 10k 3 real numbers with nonnegative sum has at least n−1 k−1 nonnegative k-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for n ≥ 4k and any weighting on the 1-dimensional subspaces of F n q with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least n−1 k−1 q . We also discuss two additional generalizations, which can be regarded as analogues of the Erdős-Ko-Rado theorem on k-intersecting families.
“…He reduced the conjecture to finding a k-uniform hypergraph on n vertices satisfying the MMS property (similar techniques were also employed earlier in [8]). The second conjecture, Conjecture 1.3, was very recently proved by Chowdhury, Sarkis, and Shahriari [3] simultaneously with our work. They also proved a quadratic bound n ≥ 8k 2 for sets.…”
An r-unform n-vertex hypergraph H is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n > 10r 3 , every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n > 10k 3 real numbers with nonnegative sum has at least n−1 k−1 nonnegative k-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for n ≥ 4k and any weighting on the 1-dimensional subspaces of F n q with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least n−1 k−1 q . We also discuss two additional generalizations, which can be regarded as analogues of the Erdős-Ko-Rado theorem on k-intersecting families.
“…A 2012 breakthrough paper by Alon, Huang, and Sudakov [1] confirmed the conjecture for n ≥ 33k 2 . More recently, Chowdhury, Sarkis, and Shahriari [6] showed that the MMS conjecture is true for n ≥ 8k 2 . A linear bound was given by Pokrovskiy [21], who proved that the MMS conjecture holds, provided that n ≥ 10 47 k.…”
In this paper we give a proof of the Manickam-Miklós-Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counterexamples to the conjecture are described.
“…Moreover, if equality holds, the family of k-element subsets with nonnegative sum is a star on x 1 , S ∈ X k : x 1 ∈ S . The proof of Theorem 1.2 is similar to that of Theorem 1.3 in [9], where we tackle the Manickam-Miklós-Singhi conjectures for sets and vector spaces simultaneously. For the reader's convenience, we present the calculations for the case of sets in full detail in this unpublished manuscript.…”
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confidence: 88%
“…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 [1,2,3,4,5,6,7,8,9,11,13,14,16,17,18,19,20,21], but still remains open. For more than two decades, Conjecture 1.1 was known to hold only when k|n [18] or when n is at least an exponential function of k [4,6,17,21].…”
More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .
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