## Maximal $$k$$-intersecting families of subsets and Boolean functions.(Russian, English)Zbl 1438.05007

Diskretn. Anal. Issled. Oper. 25, No. 4, 15-26 (2018); translation in J. Appl. Ind. Math. 12, No. 4, 797-802 (2018).
Summary: 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 $$k>2$$.

### MSC:

 05A16 Asymptotic enumeration 06E30 Boolean functions
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### References:

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