In this paper we investigate the structure of uncountable maximal antichains of Silver forcing and show that they have to be at least of size d, where d is the dominating number. Part of this work can be used to show that the additivity of the Silver forcing ideal has size at least the unbounding number b. It follows that every reasonable amoeba Silver forcing adds a dominating real.

In this paper we analyse some questions concerning trees on κ, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in [6, Question 5.2] about the diagram for regularity properties.

We present some results about the burgeoning research area concerning set theory of the “κ‐reals”. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.

We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well‐known notions, such as Lebesgue measurablility, Baire‐ and Doughnut‐property and the Marczewski field. Moreover, we prove that any absolute amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Hein's and Spinas' conjecture that.