## The number of $$k$$-undivided families of subsets of an $$n$$-element set ($$k$$-undivided Boolean functions of $$n$$-variables). III. The case when $$n$$ is arbitrary and $$k\geq 3$$.(Russian)Zbl 1249.05015

Summary: Let $$S$$ be a finite set that consists of $$n$$ different elements and $$k\geq 2$$ be a natural number. A family $$\mathcal F$$ of subsets $$S_1,\dots,S_r$$, $$r\geq k$$, of the set $$S$$ is called $$k$$-undivided if the intersection of any $$k$$ sets of $$\mathcal F$$ is non-empty. Such families are equivalent to $$k$$-undivided Boolean functions of $$n$$ variables, i.e. to functions $$f(x_1,\dots,x_n)$$ such that any $$k$$ vectors with $$f(x_1,\dots,x_n)=1$$ have at least one component equal to 1. In the paper, an asymptotics is given for the number of $$k$$-undivided Boolean functions of $$n$$ variables as $$n \to \infty$$ and $$k\geq 3$$ is fixed.
For Part I see [ibid., Ser. 1 10, No. 4, 31–69 (2003; Zbl 1032.05006)].
For Part II see [ibid., Ser. 1 12, No. 1, 12–70 (2005; Zbl 1077.05007)].

### MSC:

 05A16 Asymptotic enumeration 03E05 Other combinatorial set theory 06E30 Boolean functions

### Keywords:

two-valued function; asymptotic expression; Post class

### Citations:

Zbl 1032.05006; Zbl 1077.05007