# Borel sets with large squares

Fundamenta Mathematicae (1999)

- Volume: 159, Issue: 1, page 1-50
- ISSN: 0016-2736

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topShelah, Saharon. "Borel sets with large squares." Fundamenta Mathematicae 159.1 (1999): 1-50. <http://eudml.org/doc/212318>.

@article{Shelah1999,

abstract = { For a cardinal μ we give a sufficient condition $⊕_μ$ (involving ranks measuring existence of independent sets) for:
$⊗_μ$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a $2^\{ℵ_0\}$-square and even a perfect square,
and also for
$⊗^\{\prime \}_μ$ if $ψ ∈ L_\{ω_1, ω\}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way.
Assuming $MA + 2^\{ℵ_0\} > μ$ for transparency, those three conditions ($⊕_μ$, $⊗_μ$ and $⊗^\{\prime \}_μ$) are
equivalent, and from this we deduce that e.g. $∧_\{α < ω_1\}[ 2^\{ℵ_0\}≥ ℵ_α ⇒ ￢ ⊗_\{ℵ_α\}]$, and also that
$min\{μ: ⊗_μ\}$, if $ < 2^\{ℵ_0\}$, has cofinality $ℵ_1$.
We also deal with Borel rectangles and related model-theoretic problems.},

author = {Shelah, Saharon},

journal = {Fundamenta Mathematicae},

keywords = {Borel set; perfect square; Cohen reals; analytic set; Hanf numbers; rectangle containment; Martin's Axiom; Borel rectangles},

language = {eng},

number = {1},

pages = {1-50},

title = {Borel sets with large squares},

url = {http://eudml.org/doc/212318},

volume = {159},

year = {1999},

}

TY - JOUR

AU - Shelah, Saharon

TI - Borel sets with large squares

JO - Fundamenta Mathematicae

PY - 1999

VL - 159

IS - 1

SP - 1

EP - 50

AB - For a cardinal μ we give a sufficient condition $⊕_μ$ (involving ranks measuring existence of independent sets) for:
$⊗_μ$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a $2^{ℵ_0}$-square and even a perfect square,
and also for
$⊗^{\prime }_μ$ if $ψ ∈ L_{ω_1, ω}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way.
Assuming $MA + 2^{ℵ_0} > μ$ for transparency, those three conditions ($⊕_μ$, $⊗_μ$ and $⊗^{\prime }_μ$) are
equivalent, and from this we deduce that e.g. $∧_{α < ω_1}[ 2^{ℵ_0}≥ ℵ_α ⇒ ￢ ⊗_{ℵ_α}]$, and also that
$min{μ: ⊗_μ}$, if $ < 2^{ℵ_0}$, has cofinality $ℵ_1$.
We also deal with Borel rectangles and related model-theoretic problems.

LA - eng

KW - Borel set; perfect square; Cohen reals; analytic set; Hanf numbers; rectangle containment; Martin's Axiom; Borel rectangles

UR - http://eudml.org/doc/212318

ER -

## References

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