On the rigidity of Souslin trees and their generic branches

We show it is consistent that there is a Souslin tree S such that after forcing with S , S is Kurepa and for all clubs C ⊂ ω 1 , S ↾ C is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with ♢ that for every Souslin tree T there is a dense X ⊆ T which does not contain a copy of T . This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982).