Ramsey and Freeness Properties of Polish Planes

Suppose that X is a Polish space which is not -compact. We provethat for every Borel colouring of X² by countably many colours,there exists a monochromatic rectangle with both sides closedand not -compact. Moreover, every Borel colouring of [X]² byfinitely many colours has a homogeneous set which is closedand not -compact. We also show that every Borel measurable functionf:X² X has a free set which is closed and not -compact. Ascorollaries of the proofs we obtain two results: firstly, theproduct forcing of two copies of superperfect tree forcing doesnot add a Cohen real, and, secondly, it is consistent with ZFCto have a closed subset of the Baire space which is not -compactand has the property that, for any three of its elements, noneof them is constructible from the other two. A similar proofshows that it is consistent to have a Laver tree such that noneof its branches is constructible from any other branch. Thelast four results answer questions of Goldstern and Brendle.2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.