### Abstract

Given a set of n real numbers, if the sum of the elements of every subset of size larger than k is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is (n-1/k-1) + (n-1/k-2) + • • • + (n-1 0 )+ 1, settling a problem of Tsukerman. We provide two proofs; the first establishes and applies a weighted version of Hall's theorem, and the second is based on an extension of the nonuniform Erd.os-Ko-Rado theorem.

Original language | English (US) |
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Pages (from-to) | 811-816 |

Number of pages | 6 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Erdos-Ko-Rado
- Hall's theorem
- Nonnegative subsets

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## Cite this

Alon, N., Aydinian, H., & Huang, H. (2014). Maximizing the number of nonnegative subsets.

*SIAM Journal on Discrete Mathematics*,*28*(2), 811-816. https://doi.org/10.1137/130947295