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

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.