Uncountable trees and Cohen $\kappa$-reals

TL;DR

This paper explores amoeba concepts for tree-forcings in uncountable spaces, addressing open questions about regularity properties and establishing new relationships between Ramsey and Baire properties at uncountable cardinals.

Contribution

It generalizes previous work on amoeba for tree-forcings to uncountable spaces and answers open questions about regularity properties and their interrelations.

Findings

Established $ ext{Sigma}_1^1$-counterexamples to certain regularity properties.

Proved a strong link between Ramsey and Baire properties at uncountable cardinals.

Extended the theory of amoeba for tree-forcings beyond the standard case.

Abstract

We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show $\Sigma_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.