科研之友
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摘要
For k is an element of Z(+), let f(k) be the minimum integer N such that for all n >= N, every set of n real numbers with nonnegative sum has at least (n-1 k-1) k-element subsets whose sum is also nonnegative. In 1988, Manickam, Miklos, and Singhi proved that f (k) exists and conjectured that f(k) <= 4k. In this note, we prove f (3) = 11, f(4) <= 24, and f(5) <= 40, which improves previous upper bounds in these cases. Moreover, we show how our method could potentially yield a quadratic upper bound on f (k). We end by discussing how our methods apply to a vector space analogue of the Manickam-Miklos-Singhi conjecture.
- 出版日期2014-1
