SPM Bulletin 22

Following Jana Flaškova’s talk at this meeting, we have invited her to contribute a section to this issue. We thank her for her interesting contribution in Second 2 and in the Problem of the Issue section.

The list of problems at the end of the bulletin became longer than one page. We therefore removed the first few, and will continue this way unless some more problems are solved and their space becomes available. . . A much better version of Shelah’s paper showing that g ≤ b + is now available at arxiv.org/abs/math/0612353 Enjoy, Boaz Tsaban, boaz.tsaban@weizmann.ac.il http://www.cs.biu.ac.il/~tsaban

There have been several attempts to connect ultrafilters with families of “small” sets. Two of them -0-points and I-ultrafilters -were important for my Ph.D. thesis. The first one is due to Gryzlov [8]: an ultrafilter U ∈ N * is called a 0-point if for every one-to-one function f : N → N there exists a set U ∈ U such that f [U] has asymptotic density zero. The second term was introduced by Baumgartner [1]: Let I be a family of subsets of a set X such that I contains all singletons and is closed under subsets. Given a free ultrafilter U on N, we say that U is an I-ultrafilter if for any

In my Ph.D. thesis [3] (on which all my papers are more less based) I studied Iultrafilters in the setting X = N and I is an ideal on N or another family of “small” subsets of N that contains finite sets and is closed under subsets. As I were considered the ideal of sets with asymptotic density zero Z 0 = {A ⊆ N : lim sup n→∞ |A∩n| n = 0}, the summable ideal I 1/n = {A ⊆ N : a∈A 1 a < ∞} or the family of (almost) thin sets and (SC)-sets.

Here are the (probably not common) definitions: We say that A ⊆ N with an increasing enumeration A = {a n : n ∈ N} is thin: if lim n→∞ an a n+1 = 0; almost thin: if lim n→∞ an a n+1 < 1; (SC)-set: if lim n→∞ a n+1a n = ∞. In the thesis various examples of I-ultrafilters for all these (and also some other) families I are constructed under additional set theoretic assumptions.

In my first paper [4] it is shown that thin sets and almost thin sets actually determine the same class of I-ultrafilters and there is a proof that the existence of these ultrafilters is independent of ZFC. The relation between this class of ultrafilters and selective ultrafilters or Q-points is studied. Some construction made in the paper under CH were proved in the thesis assuming MA ctble .

The next paper [5] focuses on I-ultrafilters where I is the summable ideal I 1/n or the density ideal Z 0 . The relation between these two classes of ultrafilters is shown and also the relation to the class of P -points. Assuming CH or MA ctble several examples of these ultrafilters are constructed. Again, stronger versions of some of the results can be found in the thesis.

One of the few ZFC results in my thesis is the following: There exists an ultrafilter U ∈ N * such that for every one-to-one function f : N → N there exists a set ∈ U with f [U] in the summable ideal. This theorem strengthens Gryzlov’s result concerning the existence of 0-points and it was published also separately in [6].

Connections between various I-ultrafilters and some well-known ultrafilters such as P -points were studied in two sections of my thesis. It is known that P -points can be described as I-ultrafilters in two different ways: If X = 2 N then P -points are precisely the I-ultrafilters for I consisting of all finite and converging sequences, if X = ω 1 then P -points are precisely the I-ultrafilters for I = {A ⊆ ω 1 : A has order type ≤ ω}. My latest paper [7] deals with the question whether there is a family I of subsets of natural numbers such that P -points are precisely the I-ultrafilters. However, only some partial answers are given.

During the 1st European Set Theory Meeting in Bȩdlewo I gave a talk “On sums and products of certain I-ultrafilters”. As the title suggests it was a summary of my knowledge about sums and products of some I-ultrafilters. The slides and notes with proofs on which the talk was based (as well as my Ph.D. thesis) are available online on my webpage: http://home.zcu.cz/~flaskova Jana Flašková Department of Mathematics, University of West Bohemia 3. Research announcements 3.1. Inverse Systems and I-Favorable Spaces. Let X be a compact space. Player I has a winning strategy in the open-open game played on X if, and only if X can be represented as a limit of σ-complete inverse system of compact metrizable spaces with skeletal bonding maps.

http://arxiv.org/abs/0706.3815 Andrzej Kucharski and Szymon Plewik 3.2. Combinatorial and hybrid principles for σ-directed families of countable sets modulo finite. We consider strong combinatorial prin

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