Ideals on countable sets: a survey with questions

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions

Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

On some ideal related to the ideal (v0)

The ideal (v0) is known in the literature and is naturally linked to the structure [ω]ω. We consider some natural counterpart of the ideal (v0) related in an analogous way to the structure Dense(ℚ) and investigate its combinatorial properties. By the use of the notion of ideal type we prove that under CH this ideal is isomorphic to (v0)

Strolling through Paradise

With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S and J=R this gives a Marczewski null set which is not Ramsey, extending earlier partial results by Aniszczyk, Frankiewicz, Plewik, Brown and Corazza and answering a question of the latter. In case I=M and J=L this gives a Miller null set which is not Laver null; this answers a question addressed by Spinas. We also investigate the question which pairs of the ideals considered are orthogonal and which are not. Furthermore we include Mycielski's ideal P_2 in our discussion

Banach spaces and Ramsey Theory: some open problems

We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.Comment: 17 pages, no figures; RACSAM, to appea

Saccharinity

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal

Can ideals without ccc be interesting?

AbstractWe study those ideals I of sets in a perfect Polish space X which admit a Borel measurable f: X→X with f-1[\s{x\s}]∉I for each xϵX. A stronger version of that property (when, additionally, X is a group and I an invariant ideal) states that there exist a Borel set B∉I and a perfect P⊆X, such that \s{B + x:x ϵP\s} forms a disjoint family