# Dominating and unbounded free sets

@article{Solecki1999DominatingAU, title={Dominating and unbounded free sets}, author={Slawomir Solecki and Otmar Spinas}, journal={Journal of Symbolic Logic}, year={1999}, volume={64}, pages={75 - 80} }

Abstract We prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.

## 8 Citations

Ramsey and Freeness Properties of Polish Planes

- Mathematics
- 2001

Suppose that X is a Polish space which is not σ‐compact. We prove that for every Borel colouring of X2 by countably many colours, there exists a monochromatic rectangle with both sides closed and not…

Analytic Equivalence Relations and the Forcing Method

- PhilosophyThe Bulletin of Symbolic Logic
- 2013

Abstract I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence…

Non-meager free sets for meager relations on Polish spaces

- Mathematics
- 2013

of nowhere densesubsets of X;• is nowhere meager in X, if for any non-empty open set U⊂ Xthe intersection U∩Ais notmeager in X.It is clear that a subset A⊂ X of a Polish space X is nowhere meager if…

Canonical Behavior of Borel Functions on superperfect Rectangles

- MathematicsJ. Math. Log.
- 2001

It is shown that every Borel measurable function from (ωω)2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.

Canonical Ramsey Theory on Polish Spaces

- Mathematics
- 2013

Preface 1. Introduction 2. Background facts 3. Analytic equivalence relations and models of set theory 4. Classes of equivalence relations 5. Games and the Silver property 6. The game ideals 7.…

Mycielski among trees

- MathematicsMath. Log. Q.
- 2021

Two-dimensional version of the classical Mycielski theorem is considered by replacing a perfect square with a rectangle and a uniformly perfect tree is shown such that no side of such a rectangle can be a body of a Silver tree or a Miller tree.

On effective σ‐boundedness and σ‐compactness

- MathematicsMath. Log. Q.
- 2013

It is shown that, given a finite number of equivalence relations, any set A of the Baire space either is covered by compact sets and lightface equivalence classes of the relations, or A contains a superperfect subset which is pairwise -inequivalent for all i = 1, …, n.

Forcing Properties of Ideals of Closed Sets

- MathematicsThe Journal of Symbolic Logic
- 2011

The following dichotomy is proved: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

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