Abstract
A family of subsets of an n-element set is k-intersecting if the intersection of every k subsets in the family is nonempty. A family is maximalk-intersecting if no subset can be added to the family without violating the k-intersection property. There is a one-to-one correspondence between the families of subsets and Boolean functions defined as follows: To each family of subsets, assign the Boolean function whose unit tuples are the characteristic vectors of the subsets.We show that a family of subsets is maximal 2-intersecting if and only if the corresponding Boolean function is monotone and selfdual. Asymptotics for the number of such families is obtained. Some properties of Boolean functions corresponding to k-intersecting families are established fork > 2.
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Original Russian Text © Yu.A. Zuev, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 4, pp. 15–26.
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Zuev, Y.A. Maximal k-Intersecting Families of Subsets and Boolean Functions. J. Appl. Ind. Math. 12, 797–802 (2018). https://doi.org/10.1134/S1990478918040191
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DOI: https://doi.org/10.1134/S1990478918040191