: In this final talk of the sequence we will sketch Blinovsky's recent proof of the conjecture: Whenever n is at least 4k, and A is a set of n numbers with sum 0, then there are at least (n-1) choose (k-1) subsets of size k which have non-negative sum. The nice aspect of the proof is the combination of hypergraph concepts with convex geometry arguments and a Berry-Esseen inequality for approximating the hypergeometric distribution. The not so nice aspect (which will be omitted in the talk) is the amount of very tedious algebraic manipulation that is necessary to verify the required estimates. There are slides for all four MMS talks here.