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

TL;DR

This paper introduces a refined Mitchell-style forcing technique with highly structured model collections as side conditions, enabling the addition of simplified morasses of size  with small working parts, answering a longstanding open question.

Contribution

It presents a new Mitchell-style forcing method with small working parts and structured models, demonstrating the addition of a -sized simplified morass with conditions of size less than .

Findings

Successfully adds a (,1)-simplified morass via forcing.

Introduces a more manageable Mitchell-style forcing with structured side conditions.

Answers an open question by Shelah and Velleman from the 1980s.

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 $<\kappa$.