Ramsey and freeness properties of Polish planes (Q2766383)

Let \(X\) be a Polish space which is not \(\sigma\)-compact and consider the plane \(X^2\). The author shows that (1) for every Borel coloring of \(X^2\) by countably many colors there are \(P,Q \subseteq X\) which are closed and not \(\sigma\)-compact such that the rectangle \(P\times Q\) is monochromatic; (2) for every Borel coloring of \([X]^2\) by finitely many colors there is a homogeneous \(P \subseteq X\) which is closed and not \(\sigma\)-compact (i.e., \([P]^2\) is monochromatic); (3) for every Borel function \(f : X^2 \to X\) there is a free \(P \subseteq X\) which is closed and not \(\sigma\)-compact (i.e., \(f(x,y) \in \{ x,y \} \cup (X \setminus P)\) for all \(x,y \in P\)). NEWLINENEWLINENEWLINEThese results provide analoga to well-known results due to Galvin and Blass (which are in turn consequences of an old theorem of \textit{J. Mycielski} [Fundam. Math. 55, 139-147 (1964; Zbl 0124.01301)]) with ``\(2\)'' replaced by arbitrary ``\(n\)'' and ``closed and not \(\sigma\)-compact'' replaced by ``perfect.'' It is well known that analoga for dimension \(n \geq 3\) must fail in the ``closed and not \(\sigma\)-compact'' case. NEWLINENEWLINENEWLINESince, by Hurewicz' Theorem, a Polish space is not \(\sigma\)-compact iff it contains a homeomorphic copy of the Baire space \(\omega^\omega\), the proofs boil down to an investigation of superperfect subsets of \(\omega^\omega\). This is recast in forcing language as the study of \(Q^2\), the product of two copies of Miller's superperfect tree forcing \(Q\). The paper, then, develops intricate slow fusion arguments for products of superperfect trees as its main technical feat. For example, the main lemma leading to (1) says that given \(D \subseteq Q^2\) open dense and \((p,q) \in Q^2\), there is \((p',q') \leq (p,q)\) with the same stem such that the superperfect rectangle defined by \((p',q')\) is covered by countably many superperfect rectangles from \(D\). NEWLINENEWLINENEWLINEAs a consequence the author also obtains new results about \(Q^2\) and constructibility: namely, (4) \(Q^2\) does not add Cohen reals; (5) it is consistent there is a superperfect set \(P\) such that no member \(x\) of \(P\) is constructible from any pair of members (distinct from \(x\)) of \(P\).