## The number of $$k$$-undivided families of subsets of an $$n$$-element set ($$k$$-undivided Boolean functions of $$n$$ variables). II: The case when $$n$$ is odd and $$k=2$$.(Russian)Zbl 1077.05007

[For Part I see Diskretn. Anal. Issled. Oper., Ser. 1 10, 31–69 (2003; Zbl 1032.05006).]
Summary: Let $$S$$ be a finite set that consists of $$n$$ different elements and let $$k$$ be a natural number, $$1\leq k\leq n$$. 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 nonempty. 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 common component equal to 1. In the paper, an asymptotics is given for the size of a special subset of 2-undivided Boolean functions of $$n$$ variables (2-undivided families of subsets of an $$n$$-element set) as $$n\to\infty$$ and $$n$$ is odd. The fact that almost all 2-undivided Boolean functions of $$n$$ variables belong to the special class will be proven in the next paper.

### MSC:

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

### Keywords:

two-valued function; asymptotic expression; Post class

Zbl 1032.05006