If F ⊆ NN is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ NN. no member of which is covered by finitely many functions from F, there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality (...) condition. (shrink)

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.

We investigate cardinal invariants related to the structure of dense sets of rationals modulo the nowhere dense sets. We prove that , thus dualizing the already known [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 59–80, Theorem 3.6]. We also show the consistency of each of and . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. (...) 183 59–80, Questions 3.11]. (shrink)

A maximal almost disjoint (mad) family $\mathscr{A} \subseteq [\omega]^\omega$ is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family, A, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A], A ∈A are nowhere dense. An ℵ 0 -mad family, A, is a mad family with the property that given any countable family $\mathscr{B} \subset (...) [\omega]^\omega$ such that each element of B meets infinitely many elements of A in an infinite set there is an element of A meeting each element of B in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ 0 -mad families. Either of the conditions b = c or $\mathfrak{a} ) implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family, S, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈S are nowhere dense. Also, Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ 0 -splitting families of cardinality ≤ κ exist. (shrink)

We define a class of productive σ-ideals of subsets of the Cantor space 2 ω and observe that both σ-ideals of meagre sets and of null sets are in this class. From every productive σ-ideal I we produce a σ-ideal I κ , of subsets of the generalized Cantor space 2 κ . In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. Then we investigate additivity, (...) covering number, uniformity and cofinality of I κ . For example, we show that non(I = non(I ω 1 ) = non(I ω 2 ). Our results generalizes those from [5]. (shrink)

We investigate splitting number and reaping number for the structure (ω) ω of infinite partitions of ω. We prove that ${\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}}$ and ${\mathfrak{s}_{d}\geq\mathfrak{b}}$ . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}$ . To prove the consistency ${\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})}$ we introduce new cardinal invariants ${\mathfrak{r}_{pair}}$ and ${\mathfrak{s}_{pair}}$ . We also study the relation between ${\mathfrak{r}_{pair}, \mathfrak{s}_{pair}}$ and other cardinal invariants. We show (...) that ${\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\ma thfrak{r}}$ and ${\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})}$. (shrink)

We continue [21] and study partition numbers of partial orderings which are related to /fin. In particular, we investigate Pf, be the suborder of /fin)ω containing only filtered elements, the Mathias partial order M, and , ω the lattice of partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for Pf. We show that the partition number of is C. We also show that consistently the distributivity number of ω is smaller than (...) the distributivity number of /fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size 1 does not imply that a Polish space can be partitioned into 1 many closed sets. (shrink)