A linear bound on the Manickam-Miklos-Singhi Conjecture
Combinatorics
2013-08-12 v1 Number Theory
Abstract
Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1 \choose k-1). This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n > 33k^2. In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n > Ck.
Cite
@article{arxiv.1308.2176,
title = {A linear bound on the Manickam-Miklos-Singhi Conjecture},
author = {Alexey Pokrovskiy},
journal= {arXiv preprint arXiv:1308.2176},
year = {2013}
}
Comments
25 pages, 4 figures