First distribution invariants and EKR theorems
Journal of Combinatorial Theory Series A
A method to count the positive 3-subsets in a set of real numbers with non-negative sum
European Journal of Combinatorics
On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree
SIAM Journal on Discrete Mathematics
Bounding Probability of Small Deviation: A Fourth Moment Approach
Mathematics of Operations Research
Solution of a problem on non-negative subset sums
European Journal of Combinatorics
Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels
Journal of Combinatorial Theory Series A
A remark on the problem of nonnegative k-subset sums
Problems of Information Transmission
Improved bounds for Erdős' Matching Conjecture
Journal of Combinatorial Theory Series A
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
A note on the Manickam-Miklós-Singhi conjecture
European Journal of Combinatorics
Notes: On the number of nonnegative sums
Journal of Combinatorial Theory Series B
A linear programming approach to the Manickam-Miklós-Singhi conjecture
European Journal of Combinatorics
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More than twenty years ago, Manickam, Miklos, and Singhi conjectured that for any integers n, k satisfying n=4k, every set of n real numbers with nonnegative sum has at least (n-1k-1)k-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of 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=33k^2. This substantially improves the best previously known exponential lower bound n=e^c^k^l^o^g^l^o^g^k. In addition we prove a tight stability result showing that for every k and all sufficiently large n, every set of n reals with a nonnegative sum that does not contain a member whose sum with any other k-1 members is nonnegative, contains at least (n-1k-1)+(n-k-1k-1)-1 subsets of cardinality k with nonnegative sum.