Laver Trees in the Generalized Baire Space

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