Ramsey and Freeness Properties of Polish Planes

Suppose that X is a Polish space which is not σ‐compact. We prove that for every Borel colouring of X 2 by countably many colours, there exists a monochromatic rectangle with both sides closed and not σ‐compact. Moreover, every Borel colouring of [ X ] 2 by finitely many colours has a homogeneous set which is closed and not σ‐compact. We also show that every Borel measurable function f : X 2 → X has a free set which is closed and not σ‐compact. As corollaries of the proofs we obtain two results: firstly, the product forcing of two copies of superperfect tree forcing does not add a Cohen real, and, secondly, it is consistent with ZFC to have a closed subset of the Baire space which is not σ‐compact and has the property that, for any three of its elements, none of them is constructible from the other two. A similar proof shows that it is consistent to have a Laver tree such that none of its branches is constructible from any other branch. The last four results answer questions of Goldstern and Brendle. 2000 Mathematics Subject Classification : 03E15, 26B99, 54H05.