We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with 𝔥 > ℵ1 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.
Heike Mildenberger. "The club principle and the distributivity number." J. Symbolic Logic 76 (1) 34 - 46, March 2011. https://doi.org/10.2178/jsl/1294170988