# Symmetries between two Ramsey properties

@article{Halbeisen1998SymmetriesBT, title={Symmetries between two Ramsey properties}, author={Lorenz Halbeisen}, journal={Archive for Mathematical Logic}, year={1998}, volume={37}, pages={241-260} }

Abstract. In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet…

## 9 Citations

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## References

SHOWING 1-10 OF 34 REFERENCES

### On the Ramsey property for sets of reals

- MathematicsJournal of Symbolic Logic
- 1983

A certain two-person game is presented such that if either player has a winning strategy then a homogeneous set for the partition can be constructed, and alternative proofs of some of the known results are given.

### Partitions and filters

- MathematicsJournal of Symbolic Logic
- 1986

The author undertook to study the notion dual to that of a Ramsey ultrafilter, and establishes some basic properties of the lattice of all partitions of a cardinal κ, and introduces co-Ramsey filters, which can be associated with Ramsey ultrafilters.

### Happy families and completely Ramsey sets

- MathematicsArch. Math. Log.
- 1993

Games of Kastanas are used to obtain a new characterization of the classCℱ of all sets that are completely Ramsey with respect to a given happy family ℱ and ideas of Plewik are combined to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent of Cℽ.

### On Shattering, Splitting and Reaping Partitions

- MathematicsMath. Log. Q.
- 1998

Using some properties of the ideal 𝔍 of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, it is shown that add(𝔊) = cov(𝓂) = ℌ and it is can be shown thatℌ > ω1 is consistent with ZFC and as a corollary the authors get the relative consistency of ℍ > t, where t is the tower number.

### TOOLS FOR YOUR FORCING CONSTRUCTION

- Mathematics
- 1992

A preservation theorem is a theorem of the form: "If hP�,Q� : � < �i is an iteration of forcing notions, and every Qsatisfies ' in V P� , then Psatisfies '." We give a simplified version of a general…

### A new proof that analytic sets are Ramsey

- MathematicsJournal of Symbolic Logic
- 1974

Abstract We give a direct mathematical proof of the Mathias-Silver theorem that every analytic set is Ramsey.

### Δ 1 2 -sets of reals

- Mathematics
- 1988

Etablissement d'une chaine d'implications uniques entre quatre notions pour les ensembles de reels

### Set Theory: On the Structure of the Real Line

- Education
- 1995

This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.