  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. The best known bounds are due to Alon, Huang, and Sudakov who proved the
conjecture when n > 33k^2. In this paper we improve this bound by showing that
there is a constant C such that the conjecture holds when n > Ck.
