Properties of Laver forcing associated with a co-ideal expressed via the Katetov order

TL;DR

This paper explores variants of Laver forcing linked to co-ideals, characterizing when they add Cohen reals using the Katetov order, and investigates their combinatorial and structural properties.

Contribution

It provides a Katetov-theoretic characterization of Cohen real addition in Laver forcing from co-ideals and clarifies the relationship between Cohen reals and the Laver property.

Findings

Laver forcing from co-ideals adds Cohen reals iff characterized by Katetov order.

Such forcings never add random reals.

The addition of Cohen reals and the Laver property are not equivalent, even for ultrafilters.

Abstract

We study variants of classical Laver forcing defined from co-ideals and analyze their combinatorial properties in terms of the Kat\v{e}tov order. In particular, we give a Kat\v{e}tov-theoretic characterization of when Laver forcing associated with a co-ideal adds Cohen reals, and we show that such forcings never add random reals. Improving a result of B{\l}aszczyk and Shelah we prove that the addition of Cohen reals and the Laver property are not equivalent, even in the case of ultrafilters. As an application, we investigate the problem of adding half Cohen reals without adding Cohen reals via tree-like forcings arising from co-ideals. We obtain several partial results and structural obstructions. Finally, we resolve several open questions from the literature concerning the one-to-one or constant property and cardinal invariants associated with ideals.