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

Abstract: We study variants of classical Laver forcing defined from co-ideals and analyze their combinatorial properties in terms of the Katětov order. In particular, we give a Katětov-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 Błaszczyk 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.