A New Quadratic Bound for the Manickam-Mikl\'os-Singhi Conjecture
A Chowdhury, G Sarkis, S Shahriari - arXiv preprint arXiv:1403.1844, 2014 - arxiv.org
A Chowdhury, G Sarkis, S Shahriari
arXiv preprint arXiv:1403.1844, 2014•arxiv.orgMore than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive
integers $ n, k $ with $ n\geq 4k $, every set of $ n $ real numbers with nonnegative sum has
at least $\binom {n-1}{k-1} $$ k $-element subsets whose sum is also nonnegative. We verify
this conjecture when $ n\geq 8k^ 2$, which simultaneously improves and simplifies a bound
of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $ k< 10^{45} $.
integers $ n, k $ with $ n\geq 4k $, every set of $ n $ real numbers with nonnegative sum has
at least $\binom {n-1}{k-1} $$ k $-element subsets whose sum is also nonnegative. We verify
this conjecture when $ n\geq 8k^ 2$, which simultaneously improves and simplifies a bound
of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $ k< 10^{45} $.
More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers with , every set of real numbers with nonnegative sum has at least -element subsets whose sum is also nonnegative. We verify this conjecture when , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when .
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