Laver Trees in the Generalized Baire Space

TL;DR

This paper investigates generalized Laver forcing in uncountable Baire spaces, showing it necessarily adds Cohen reals and establishing conditions under which certain tree forcings add Cohen reals, thus advancing understanding of these spaces.

Contribution

It demonstrates that generalized Laver forcing adds Cohen reals and characterizes when certain tree forcings add Cohen reals in generalized Baire spaces.

Findings

Generalized Laver forcing adds Cohen $oldsymbol{ ext{kappa}}$-real.

A dichotomy and ideal related to generalized Laver forcing are studied.

Certain $<oldsymbol{ ext{kappa}}$-distributive tree forcings add Cohen $oldsymbol{ ext{kappa}}$-real.

Abstract

We prove that any suitable generalization of Laver forcing to the space $ \kappa^\kappa$, for uncountable regular $\kappa$, necessarily adds a Cohen $\kappa$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ \kappa^{<\kappa}=\kappa$, then every $<\kappa$-distributive tree forcing on $\kappa^\kappa$ adding a dominating $\kappa$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $\kappa$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140