# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Borel sets with large squares

### Tom 159 / 1999

Fundamenta Mathematicae 159 (1999), 1-50 DOI: 10.4064/fm-159-1-1-50

#### Streszczenie

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 $⊗'_μ$ 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 $⊗'_μ$) 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.

#### Autorzy

• Saharon Shelah

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