An improved bound for the Manickam-Miklós-Singhi conjecture

2010/11/12 by Mykhaylo Tyomkyn, Tyomkyn, Mykhaylo
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #math.CO

paper · pdf · doi:10.48550/arxiv.1011.2803

arxiv created 2010/11/12 · openalex publication_date 2010/11/12 · arxiv updated 2010/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for n>k(4elog k)^k every set \x₁,..., x_n\ of n real numbers with ∑_i=0ⁿx_i ≥ 0 has at least \binomn-1k-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+k²)+k, proved by Manickam and Miklós \citeMM in 1987.

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