We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

 For a cardinal μ we give a sufficient condition ${\oplus }_{\mu }$ (involving ranks measuring existence of independent sets) for: ${\otimes }_{\mu }$ 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 $2^{{\aleph }_0}$-square and even a perfect square, and also for ${\otimes }_{\mu }^{\text{'}}$ if $\psi \in L_{{\omega }_1,\omega }$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $MA+2^{{\aleph }_0}>\mu$ for transparency, those three conditions (${\oplus }_{\mu }$, ${\otimes }_{\mu }$ and ${\otimes }_{\mu }^{\text{'}}$) are equivalent, and from this we deduce that...