Almost disjoint refinements and mixing reals

Abstract

We investigate families of subsets of ω with almost disjoint refinements in the classical case as well as with respect to given ideals on ω. More precisely, we study the following topics and questions:1) Examples of projective ideals.2) We prove the following generalization of a result due to J. Brendle:3) The existence of perfect I-almost disjoint (I-AD) families; and the existence of a "nice" ideal I on ω with the property: Every I-AD family is countable but I is nowhere maximal.4) The existence of (I, Fin)-almost disjoint refinements of families of Ipositive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We show that under Martin's Axiom if I is an everywhere meager ideal and H ⊆ I + with |H| < c, then H has an (I, Fin)-ADR, that is, a family {A H : H ∈ H} such that (i) A H ⊆ H, A H ∈ I + for every H and (ii) A H 0 ∩ A H 1 is finite for every distinct H 0 , H 1 ∈ H. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function f :This property is relevant concerning almost disjoint refinements because it is very easy to find an almost disjoint refinement of [ω] ω ∩ V in every extension V ⊆ W containing a mixing injection over V .2010 Mathematics Subject Classification. 03E05,03E15,03E35.

Ideals on countable sets: a survey with questions

“…Then I wf is also a complete co-analytic ideal. In sections 6 and 9 we shall present another examples of co-analytic ideals (see also [16,43]).…”

Towers in Filters, Cardinal Invariants, and Luzin Type Families

“…Theorem 2.5. (see [26]) For every natural number n > 0, there are Σ ∼ There is a natural way of defining nice ideals on ω. A function ϕ : P(ω) → [0, ∞] is a submeasure on ω if ϕ( ) = 0; ϕ(X ) ≤ ϕ(X ∪ Y ) ≤ ϕ(X ) + ϕ(Y ) for every X , Y ⊆ ω; and ϕ({n}) < ∞ for every n ∈ ω. ϕ is lower semicontinuous (lsc, in short) if ϕ(X ) = lim n→∞ ϕ(X ∩ n) for each X ⊆ ω. ϕ is finite if ϕ(ω) < ∞.…”

“…Then the ideal on < generated by {I x : x ∈ } is Σ ∼ 1 1 -complete. Theorem 2.3 (see [27]). The ideal of graphs without infinite complete subgraphs,…”

“…is Π ∼ 1 1 -complete (in P([ ] 2 )), tall, and a non P-ideal. Theorem 2.4 (see [27]). The ideal A ⊆ × :…”

“…is Π ∼ 1 1 -complete (in P( × )), tall, and a non P-ideal. Theorem 2.5 (see [27]). For every natural number n > 0, there are Σ ∼ 1 n -and Π ∼ 1 n -complete tall ideals on .…”