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 n > 10r3, 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 n > 10k3 real numbers with nonnegative sum has at least (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 ≥ 4k and any weighting on the 1-dimensional subspaces of Fnq with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least [n-1 k-1]q. We also discuss two additional generalizations, which can be regarded as analogues of the Erdos-Ko-Rado theorem on k-intersecting families.
|Original language||English (US)|
|Journal||Electronic Journal of Combinatorics|
|State||Published - Jul 10 2014|