Abstract
Lower and upper bounds are obtained for the size ζ(n, r, s, k) of a minimum system of common representatives for a system of families of k-element sets. By ζ(n, r, s, k) wemean themaximum (over all systems Σ = {M1, …, Mr} of sets Mi consisting of at least s subsets of {1, …, n} of cardinality not exceeding k) of the minimum size of a system of common representatives of Σ. The obtained results generalize previous estimates of ζ(n, r, s, 1).
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Funding
This work was supported by the Russian Foundation for Basic Research (grant 18-01-00355) and by the Presidential Program for the State Support of Leading Scientific Schools (grant NSh-6760.2018.1).
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 3, pp. 387–394.
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Kovalenko, K.D., Raigorodsky, A.M. Systems of Representatives. Math Notes 106, 372–377 (2019). https://doi.org/10.1134/S0001434619090062
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DOI: https://doi.org/10.1134/S0001434619090062