00-4   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Ramsey and Freeness Properties of Polish Planes

Suppose that $X$ is a Polish space which is not $\sigma$-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 $\sigma$-compact. Moreover, every Borel colouring of $[X]^2$ by finitely many colours has a homogeneous set which is closed and not $\sigma$-compact. We also show that every Borel measurable function $f:X^2 \rightarrow X$ has a free set which is closed and not $\sigma$-compact. As corollaries of the proofs we obtain that, firstly, the product forcing of two copies of superperfect tree forcing does not add a Cohen real, and, secondly, that it is consistent with ZFC to have a closed subset of the Baire space which is not $\sigma$-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 (see [2, Questions 6.7, 6.8, 6.10]).

Mathematics Subject Classification (1991): 03E15, 26B99, 54H05

Bibliographical note: Proceedings of the London Mathematical Society (3) 82 (2001), 31-63.

Keywords: Polish space, Baire space, Borel function, homogeneous set, free set, Cohen forcing, Miller forcing, Laver forcing, constructibility