We present several models which satisfy CH and some ♦-like principles while others fail, answering a question of Moore, Hrušák and Džamonja.

We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on Pω (λ) with the partition property for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.

We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with h > N₁ is consistent. We work with a class of axiom A forcings with countable conditions such that q ≥ n p is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for (...) forcings with creatures that are translated into trees. Both lead to new models of the club principle. (shrink)