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 Diskretn. Anal. Issled. Oper.: Year: Volume: Issue: Page: Find

 Diskretn. Anal. Issled. Oper., 2018, Volume 25, Number 4, Pages 15–26 (Mi da906)

Maximal $k$-intersecting families of subsets and Boolean functions

Yu. A. Zuev

Bauman Moscow State Technical University, 5 Vtoraya Baumanskaya St., 105005 Moscow, Russia

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 maximal $k$-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 for $k>2$. Bibliogr. 9.

Keywords: k-intersecting family of subsets, monotone selfdual Boolean function, layer of Boolean cube.

DOI: https://doi.org/10.17377/daio.2018.25.602

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English version:
Journal of Applied and Industrial Mathematics, 2018, 12:4, 797–802

Document Type: Article
UDC: 519.71
Revised: 16.03.2018

Citation: Yu. A. Zuev, “Maximal $k$-intersecting families of subsets and Boolean functions”, Diskretn. Anal. Issled. Oper., 25:4 (2018), 15–26; J. Appl. Industr. Math., 12:4 (2018), 797–802

Citation in format AMSBIB
\Bibitem{Zue18} \by Yu.~A.~Zuev \paper Maximal $k$-intersecting families of subsets and Boolean functions \jour Diskretn. Anal. Issled. Oper. \yr 2018 \vol 25 \issue 4 \pages 15--26 \mathnet{http://mi.mathnet.ru/da906} \crossref{https://doi.org/10.17377/daio.2018.25.602} \transl \jour J. Appl. Industr. Math. \yr 2018 \vol 12 \issue 4 \pages 797--802 \crossref{https://doi.org/10.1134/S1990478918040191} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85058073919}