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)

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