Mycielski among trees

Abstract

Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1{]}^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider generalizations of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of other types of trees with $A\subseteq B$. In particular, we show that for every comeager $G_{\delta }$ set $G\subseteq {\omega }^{\omega }\times {\omega }^{\omega }$ there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cup \Delta$ and that $P$ cannot be a Miller tree. In the case of measure we show that for every subset $F$ of $2^{\omega }\times 2^{\omega }$ of full measure there exists a uniformly perfect tree $P\subseteq 2^{<\omega }$ such that $[P]\times [P]\subseteq F\cup \Delta$ and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions of the real line from which we derive nonstandard proofs of Mycielski-like theorems via Shoenfield Absoluteness Theorem.