More than twenty years ago, Manickam, Miklos, and Singhi conjectured that for any integers n ≥ 4k, every set of n real numbers with nonnegative sum has at least binom n-1k-1 k-element subsets whose sum is also nonnegative.
In this talk we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of Feige and Samuels on estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k2. This substantially improves the best previously known exponential lower bound on n. If times permits we also mention application of our technique to several other problems.
Joint work with N. Alon and H. Huang.