Abstract
Given a set of n real numbers with a nonnegative sum, consider the family of all its k-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in [1]. The latter, in a special case, is nothing else but the problem of determining a minimal number c n (k) such that any k-uniform hypergraph on n vertices having c n (k) + 1 edges has a perfect fractional matching. We show that results obtained in [1] can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution.
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Original Russian Text © H. Aydinian, V.M. Blinovsky, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 4, pp. 56–61.
Supported in part by the DFG, project no. AH46/7-1 “General Theory of Information Transfer.”
Supported in part by the Bundesministerium für Bildung und Forschung, Germany, grant no. Q1BQ1052.
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Aydinian, H., Blinovsky, V.M. A remark on the problem of nonnegative k-subset sums. Probl Inf Transm 48, 347–351 (2012). https://doi.org/10.1134/S0032946012040059
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DOI: https://doi.org/10.1134/S0032946012040059