We show that a $$ -simplified morass can be added by a forcing with working parts of size smaller than $\kappa $. This answers affirmatively the question, asked independently by Shelah and Velleman in the early 1990s, of whether it is possible to do so.Our argument use a modification of a technique of Mitchell’s for adding objects of size $\omega _2$ in which collections of models – all of equal, countable size – are used as side conditions. In our modification, (...) whilst the individual models are, as in Mitchell’s technique, taken ad hoc from quite general classes, the collections of models are very highly structured, in a way that is somewhat different from, perhaps more stringent than, Mitchell’s original, arguably making the method more wieldy and giving the prospect of further uses with more delicate working parts. (shrink)

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The paper is (...) dedicated to the memory of Jim Baumgartner whose seminal joint paper :109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \ with the appropriate \-version of property \ for regular \ satisfying \. (shrink)

We further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω 1 , 1)-morass, then there exists a ccc forcing of size ω 1 that adds an ω 2 -Suslin tree. (2) If there is a simplified (ω 1 , 2)-morass, then there exists a ccc forcing of size ω 1 that adds a 0-dimensional Hausdorff topology τ on ω 3 which has spread (...) s(τ) = ω 1 . While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserve GCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈X α | α < ω 2 〉 such that X α ⊆ ω 1 , X β — X α is finite and X α — X β has size ω 1 for all β < α < ω 2. (shrink)