# Borel sets with large squares

@article{Shelah1998BorelSW, title={Borel sets with large squares}, author={Saharon Shelah}, journal={arXiv: Logic}, year={1998} }

For a cardinal mu we give a sufficient condition (*)_mu (involving ranks measuring existence of independent sets) for:
[(**)_mu] if a Borel set B subseteq R x R contains a mu-square (i.e. a set of the form A x A, |A|= mu) then it contains a 2^{aleph_0}-square and even a perfect square,
and also for
[(***)_mu] if psi in L_{omega_1, omega} has a model of cardinality mu then it has a model of cardinality continuum generated in a nice, absolute way.
Assuming MA + 2^{aleph_0}> mu for… Expand

#### 36 Citations

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- Notre Dame J. Formal Log.
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- Mathematics, Computer Science
- Math. Log. Q.
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It is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. Expand

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- Mathematics
- 2002

A coloring of a set X is any subset C of [X] N , where N > 1 is a natural number. We give some sufficient conditions for the existence of a perfect C-homogeneous set, in the case where C is G δ and X… Expand

Borel sets without perfectly many overlapping translations

- Mathematics
- 2018

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space
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- Mathematics, Computer Science
- Notre Dame J. Formal Log.
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The set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products, and it is proved that if $\aleph_\alpha$ is characterized byA Scott sentence, at least one of $\ aleph_alpha$ and $ is homogeneously characterizable. Expand

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We consider the following dichotomy for $\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa$: either all $R$-independent sets are of size at most… Expand

Perfect cliques and G colorings of Polish spaces

- Mathematics
- 2001

A coloring of a set X is any subset C of [X] N , where N > 1 is a natural number. We give some sucient conditions for the existence of a perfect C-homogeneous set, in case where C is G and X is a… Expand

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- 2008

Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion… Expand

Andrzej ROS LANOWSKI and Saharon SHELAH BOREL SETS WITHOUT PERFECTLY MANY OVERLAPPING TRANSLATIONS

- 2019

A b s t r a c t. We study the existence of Borel sets B ⊆ 2 admitting a sequence 〈ηα : α < λ〉 of distinct elements of 2 such that ∣(ηα+B)∩ (ηβ +B) ∣∣ ≥ 6 for all α, β < λ but with no perfect set of… Expand

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