An improved bound for the Manickam-Miklós-Singhi conjecture

Mathematics – Combinatorics

Scientific paper

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Scientific paper

We show that for $n>k(4e\log k)^k$ every set $\{x_1,..., x_n\}$ of $n$ real
numbers with $\sum_{i=0}^{n}x_i \geq 0$ has at least $\binom{n-1}{k-1}$
$k$-element subsets of a non-negative sum. This is a substantial improvement on
the best previously known bound of $n>(k-1)(k^k+k^2)+k$, proved by Manickam and
Mikl\'os \cite{MM} in 1987.

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