An $r$-uniform $n$-vertex hypergraph $H$ is said to have the Manickam-Miklós-Singhi?(MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of $H$. In this paper we show that for $n10r^3$, every $r$-uniform $n$-vertex hypergraph with equal codegrees has the MMS property, and the bound on $n$ is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of $n10k^3$ real numbers with nonnegative sum has at least $inom{n-1}{k-1}$ nonnegative $k$-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi?conjecture which states that for $n ge 4k$ and any weighting on the $1$-dimensional subspaces of $mathbb{F}_{q}^n$ with nonnegative sum, the number of nonnegative $k$-dimensional subspaces is at least ${n-1 rack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erd?s-Ko-Rado?theorem on $k$-intersecting families.
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机译：假设$ r $一致的$ n $顶点超图$ H $具有Manickam-Miklós-Singhi？（MMS）属性，如果对每个具有非负和的顶点权重分配，其总权重的边数非负数至少是$ H $的最小度数。在本文中，我们表明，对于$ n> 10r ^ 3 $，每个具有相同码数的$ r $-均匀$ n $-顶点超图都具有MMS属性，并且在$ n $上的界线基本上严格到一个恒定因子。这个结果有两个直接的推论。首先，它显示每组具有非负和的$ n> 10k ^ 3 $实数至少具有$ binom {n-1} {k-1} $非负$ k $-和，验证了Manickam-Miklós-Singhi这个范围的猜想。更重要的是，它暗示了向量空间Manickam-Miklós-Singhi？猜想，其中指出$ n ge 4k $和$ mathbb {F} _ {q} ^ n $的$ 1 $维子空间上的任何权重为非负和，非负$ k $维子空间的数量至少为$ {n-1 brack k-1} _q $。我们还将讨论另外两个概化，它们可以看作是相交的kk $家庭的Erd？s-Ko-Rado？定理的类似物。
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