Theorems on sets not belonging to algebras

Let $\mathcal {A}_1,\dots , \mathcal {A}_n, \mathcal {A}_{n+1}$ be a finite sequence of algebras of sets given on a set $X$, $\bigcup _{k=1}^n \mathcal {A}_k \ne \mathfrak {P}(X)$, with more than $\frac {4}{3}n$ pairwise disjoint sets not belonging to $\mathcal {A}_{n+1}$. It has been shown in the author’s previous articles that in this case $\bigcup _{k=1}^{n+1} \mathcal {A}_k \ne \mathfrak {P}(X)$. Let us consider, instead of $\mathcal {A}_{n+1}$, a finite sequence of algebras $\mathcal {A}_{n+1}, \dots , \mathcal {A}_{n+l}$. It turns out that if for each natural $i \le l$ there exist no less than $\frac {4}{3}(n+l)- \frac {l}{24}$ pairwise disjoint sets not belonging to $\mathcal {A}_{n+i}$, then $\bigcup _{k=1}^{n+l} \mathcal {A}_k \ne \mathfrak {P}(X)$. Besides this result, the article contains: an essentially important theorem on a countable sequence of almost $\sigma$-algebras (the concept of almost $\sigma$-algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function $\mathfrak {v}(n)$ that was introduced by the author in 2002, and other new results.