201308.2176v1 A linear bound on the Manickam-Miklos-Singhi Conjecture

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. Th..