Families of sets not belonging to algebras and combinatorics of finite sets of ultrafilters

Abstract

This article is a part of the theory developed by the author in which the following problem is solved under natural assumptions: to find necessary and sufficient conditions under which the union of at most countable family of algebras on a certain set X is equal to \(\mathcal{P}(X)\). Here the following new result is proved. Let \(\{\mathcal{A}_{\lambda }\}_{\lambda \in \Lambda }\) be a finite collection of algebras of sets given on a set X with \(\# (\Lambda ) =n>0\), and for each λ there exist at least \(\frac{10}{3}n+\sqrt{\frac{2n}{3}}\) pairwise disjoint sets belonging to \(\mathcal{P}(X)\setminus\mathcal{A}_{\lambda }\). Then there exists a family \(\{U^{1}_{\lambda }, U^{2}_{\lambda }\}_{\lambda \in \Lambda }\) of pairwise disjoint subsets of X (\(U^{i}_{\lambda }\cap U^{j}_{\lambda '}=\emptyset\) except the case \(\lambda =\lambda '\), \(i=j\)); and for each λ the following holds: if \(Q\in \mathcal{P}(X)\) and Q contains one of the two sets \(U^{1}_{\lambda }\), \(U^{2}_{\lambda }\), and its intersection with the other set is empty, then \(Q\notin \mathcal{A}_{\lambda }\).