Forcing over a free Suslin tree
John Krueger, Šárka Stejskalová · Advances in Mathematics · 2026
We introduce a forcing for adding almost disjoint automorphisms of a normal infinitely splitting 𝜔 1 -tree T with countable approximations. Assuming that T is a free Suslin tree, this forcing is totally proper, preserves the Suslinness of T , and does not add new cofinal branches of 𝜔 1 -trees existing in intermediate extensions. If κ is an inaccessible cardinal, then the product of the automorphism forcing of length κ with the Lévy collapse of κ to become 𝜔 2 forces that there exists an almost Kurepa Suslin tree and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Jin, Shelah, and Moore.