### 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-1k-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 vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S⊂. V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n≥. 3. k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

Original language | English (US) |
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Pages (from-to) | 84-103 |

Number of pages | 20 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 128 |

Issue number | 1 |

DOIs | |

State | Published - Nov 2014 |

### Keywords

- Erdos-Ko-Rado type theorems
- Manickam-Miklós-Singhi conjecture
- Nonnegative sums
- Vector space analogues

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## Cite this

*Journal of Combinatorial Theory. Series A*,

*128*(1), 84-103. https://doi.org/10.1016/j.jcta.2014.07.004