Mitchell-style forcing, with small working parts and collections of models as side conditions, and gap-one simplified morasses

Abstract

We give a modification of Mitchell's technique for adding objects of size $\omega_2$ with conditions with finite working parts in which the collections of models used as side conditions are very highly structured, arguably making them more wieldy. We use one such forcing (essentially a `pure side conditions' forcing) to answer affirmatively the question, asked independently by Shelah and Velleman in the late 1980s, as to whether a $(\kappa^+,1)$-simplified morass can be added by a forcing with working parts of size $<\kappa$.