Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $, into the Esterle algebra of formal power series. (...) Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on X. (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)

This paper concerns the theory of morasses. In the early 1970s Jensen defined (κ,α)-morasses for uncountable regular cardinals κ and ordinals $\alpha . In the early 1980s Velleman defined (κ, 1)-simplified morasses for all regular cardinals κ. He showed that there is a (κ, 1)-simplified morass if and only if there is (κ, 1)-morass. More recently he defined (κ, 2)-simplified morasses and Jensen was able to show that if there is a (κ, 2)-morass then there is a (κ, 2)-simplified morass. (...) In this paper we prove the converse of Jensen's result, i.e., that if there is a (κ, 2)-simplified morass then there is a (κ, 2)-morass. (shrink)

We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.

We prove that in the Cohen extension adding ℵ3 generic reals to a model of containing a simplified (ω1, 2)‐morass, gap‐2 morass‐definable η1‐orderings with cardinality ℵ3 are order‐isomorphic. Hence it is consistent that and that morass‐definable η1‐orderings with cardinality of the continuum are order‐isomorphic. We prove that there are ultrapowers of over ω that are gap‐2 morass‐definable. The constructions use a simplified gap‐2 morass, and commutativity with morass‐maps and morass‐embeddings, to extend a transfinite back‐and‐forth construction of order‐type ω1 to an (...) order‐preserving bijection between objects of cardinality ℵ3. (shrink)

The paper discusses various relationships between the concepts mentioned in the title. In Section 1 Todorcevic functions are shown to arise from both morasses and square. In Section 2 the theme is of supplements to morasses which have some of the flavour of square. Distinctions are drawn between differing concepts. In Section 3 forcing axioms related to the ideas in Section 2 are discussed.