Canonical behavior of Borel functions on superperfect rectangles (Q2772964)

The present paper is a contribution to ``Canonical Ramsey Theory'', and has its roots in the work of \textit{H. J. Prömel} and \textit{B. Voigt} [J. Comb. Theory, Ser. A 40, 409-417 (1985; Zbl 0645.05013)]. \textit{H. Lefmann} [in: Irregularities of partitions, Pap. Meet., Fertod/Hung. 1986, Algorithms Comb. 8, 93-105 (1989; Zbl 0686.05004)] proved that for every \(n\) there exists a finite set \((f^n_i)_{i < h(n)}\) of Baire measurable functions \([2^\omega]^n \to \mathbb R\) such that for every Baire measurable \(g: [2^\omega]^n \to \mathbb R\) there exists \(i < h(n)\) and a perfect set \(P \subseteq 2^\omega\) satisfying \(f^n_i (\bar{x}) = g(\bar{x})\) for every \(\bar{x} \in [P]^n\). In this situation we say that \(f^n_i\) canonizes \(g\) on a perfect square. NEWLINENEWLINENEWLINEIn this paper, canonization is transposed from the Cantor space to the Baire space, and this leads naturally to substitute perfect sets with superperfect sets (which are homeomorphic to \(\omega^\omega\)). However, it can be shown that Baire measurable functions do not have canonizations in the new context, and the author replaces them by Borel measurable functions. Another limitation comes from dimension, and in dimension \(3\) and above no positive result seems to be possible. NEWLINENEWLINENEWLINEThe paper contains a proof of the following theorem: If \(f: (\omega^\omega)^2 \to \mathbb R\) is Borel, there exists a superperfect rectangle \(S_1 \times S_2 \subseteq (\omega^\omega)^2\) (i.e.\ \(S_1\) and \(S_2\) are both superperfect) on which \(f\) is either one-to-one in at least one coordinate or else constant. This theorem is then subsumed by a more general result which provides the canonization for Borel functions. The proofs rely heavily on the analysis of Miller forcing carried out by Spinas himself in a previous paper [Proc. Lond. Math. Soc., III. Ser. 82, No. 1, 31-63 (2001; Zbl 1020.03042)].