Abstract
In this paper we prove the Ahlswede-Khachatrian conjecture [1] up to a finite number of cases, which can be checked using modern computers. This conjecture implies the conjecture from [2] and the Manickam-Miklós-Singhi conjecture.
Similar content being viewed by others
References
Ahlswede, R. and Khachatrian, L., Cone Dependence—A Basic Combinatorial Concept, Des. Codes Cryptogr., 2003, vol. 29, no. 1–3, pp. 29–40.
Aydinian, H. and Blinovsky, V.M., A Remark on the Problem of Nonnegative k-Subset Sums, Probl. Peredachi Inf., 2012, vol. 48, no. 4, pp. 56–61 [Probl. Inf. Trans. (Engl. Transl.), 2012, vol. 48, no. 4, pp. 347–351].
Bier, T., A Distribution Invariant for Association Schemes and Strongly Regular Graphs, Linear Algebra Appl., 1984, vol. 57, pp. 105–113.
Bier, T. and Manickam, N., The First Distribution Invariant of the Johnson-Scheme, Southeast Asian Bull. Math., 1987, vol. 11, no. 1–2, pp. 61–68.
Manickam, N. and Miklós, D., On the Number of Nonnegative Partial Sums of a Nonnegative Sum, Combinatorics (Proc. 7th Hung. Colloq. Eger, Hungary, July 5–10, 1987), Hajnal, A., Locász, L., and Sós, V.T., Eds., Colloq. Math. Soc. János Bolyai, vol. 52, Amsterdam: North-Holland, 1988, pp. 385–392.
Manickam, N. and Singhi, N.M., First Distribution Invariants and EKR Theorems, J. Combin. Theory, Ser. A, 1988, vol. 48, no. 1, pp. 91–103.
Alon, N., Huang, H., and Sudakov, B., Nonnegative k-Sums, Fractional Covers, and Probability of Small Deviations, J. Combin. Theory, Ser. B, 2012, vol. 102, no. 3, pp. 784–796.
Pokrovskiy, A., A Linear Bound on the Manickam-Miklós-Singhi Conjecture, arXiv:1308.2176 [math.CO], 2013.
Baranyai, Zs., On the Factorization of the Complete Uniform Hypergraph, Infinite and Finite Sets: To Paul Erdős on His 60th Birthday (Proc. Colloq. on Infinite and Finite Sets, Keszthely, Hungary, June 25–July 1, 1973), Hajnal, A., Rado, R., and Soś, V.T., Eds., Colloq. Math. Soc. János Bolyai, vol. 10, Amsterdam: North-Holland, 1975, pp. 91–108.
Chowdhury, A., A Note on the Manickam-Mikloś-Singhi Conjecture, European J. Combin., 2014, vol. 35, pp. 131–140.
Lahiri, S.N. and Chatterjee, A., A Berry-Esseen Theorem for Hypergeometric Probabilities under Minimal Conditions, Proc. Amer. Math. Soc., 2007, vol. 135, no. 5, pp. 1535–1545.
Shevtsova, I., On the Absolute Constants in the Berry-Esseen Type Inequalities for Identically Distributed Summands, arXiv:1111.6554 [math.PR], 2011.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.M. Blinovsky, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 4, pp. 43–54.
Supported by the São Paulo Research Foundation (FAPESP), project nos. 2012/13341-8 and 2013/07699-0, and NUMEC/USP, Project MaCLinC-USP.
Rights and permissions
About this article
Cite this article
Blinovsky, V.M. Minimum number of edges in a hypergraph guaranteeing a perfect fractional matching and the MMS conjecture. Probl Inf Transm 50, 340–349 (2014). https://doi.org/10.1134/S0032946014040048
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946014040048