Geodesic
Maximal $k$-intersecting families of subsets and Boolean functions
Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 4, pp. 15-26

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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. 
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     author = {Yu. A. Zuev},
     title = {Maximal $k$-intersecting families of subsets and {Boolean} functions},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {15--26},
     publisher = {mathdoc},
     volume = {25},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2018_25_4_a1/}
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Yu. A. Zuev. Maximal $k$-intersecting families of subsets and Boolean functions. Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 4, pp. 15-26. http://geodesic.mathdoc.fr/item/DA_2018_25_4_a1/