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Perfect set theorems

Otmar Spinas (2008)

Fundamenta Mathematicae

We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for G δ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

Saharon Shelah, R. Jin (1992)

Fundamenta Mathematicae

By an ω 1 - tree we mean a tree of power ω 1 and height ω 1 . Under CH and 2 ω 1 > ω 2 we call an ω 1 -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between ω 1 and 2 ω 1 . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2 ω 1 > ω 2 that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus 2 ω 1 = ω 4 that there only exist Kurepa trees with ω 3 -many branches, which answers another...

Potential isomorphism and semi-proper trees

Alex Hellsten, Tapani Hyttinen, Saharon Shelah (2002)

Fundamenta Mathematicae

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. ...

Prediction problems and ultrafilters on ω

Alan D. Taylor (2012)

Fundamenta Mathematicae

We consider prediction problems in which each of a countably infinite set of agents tries to guess his own hat color based on the colors of the hats worn by the agents he can see, where who can see whom is specified by a graph V on ω. Our interest is in the case in which 𝓤 is an ultrafilter on the set of agents, and we seek conditions on 𝓤 and V ensuring the existence of a strategy such that the set of agents guessing correctly is of 𝓤-measure one. A natural necessary condition is the absence...

Prenormality of ideals and completeness of their quotient algebras

A. Morawiec, B. Węglorz (1993)

Colloquium Mathematicae

It is well known that if a nontrivial ideal ℑ on κ is normal, its quotient Boolean algebra P(κ)/ℑ is κ + -complete. It is also known that such completeness of the quotient does not characterize normality, since P(κ)/ℑ turns out to be κ + -complete whenever ℑ is prenormal, i.e. whenever there exists a minimal ℑ-measurable function in κ κ . Recently, it has been established by Zrotowski (see [Z1], [CWZ] and [Z2]) that for non-Mahlo κ, not only is the above condition sufficient but also necessary for P(κ)/ℑ...

Questions

Alexey Ostrovsky (2005)

Acta Universitatis Carolinae. Mathematica et Physica

Rainbow Ramsey theorems for colorings establishing negative partition relations

András Hajnal (2008)

Fundamenta Mathematicae

Given a function f, a subset of its domain is a rainbow subset for f if f is one-to-one on it. We start with an old Erdős problem: Assume f is a coloring of the pairs of ω₁ with three colors such that every subset A of ω₁ of size ω₁ contains a pair of each color. Does there exist a rainbow triangle? We investigate rainbow problems and results of this style for colorings of pairs establishing negative "square bracket" relations.

Ramseyan ultrafilters

Lorenz Halbeisen (2001)

Fundamenta Mathematicae

We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...

Ramsey-type theorems

Gavalec, Martin, Vojtáš, Peter (1980)

Abstracta. 8th Winter School on Abstract Analysis

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