Nonnegative k-sums, fractional covers, and probability of small deviations

Noga Alon*, Hao Huang, Benny Sudakov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


More than twenty years ago, Manickam, Miklós, 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 ckloglogk. 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.

Original languageEnglish
Pages (from-to)784-796
Number of pages13
JournalJournal of Combinatorial Theory. Series B
Issue number3
StatePublished - May 2012


FundersFunder number
National Science Foundation226718, DMS-0812005, 1101185, DMS-1101185
Engineering Research Centers
United States-Israel Binational Science Foundation


    • Fractional cover
    • Hypergraph matching
    • Nonnegative k-sum


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