An improved bound for the Manickam–Miklós–Singhi conjecture
We show that for n > k 2 ( 4 e log k ) k , every set { x 1 , ⋯ , x n } of n real numbers with ∑ i = 1 n x i ≥ 0 has at least ( 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 , prove...
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| Veröffentlicht in: | European journal of combinatorics 2012, Vol.33 (1), p.27-32 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We show that for
n
>
k
2
(
4
e
log
k
)
k
, every set
{
x
1
,
⋯
,
x
n
}
of
n
real numbers with
∑
i
=
1
n
x
i
≥
0
has at least
(
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 Miklós
[9] in 1987. |
|---|---|
| ISSN: | 0195-6698 1095-9971 |
| DOI: | 10.1016/j.ejc.2011.07.006 |