Colouring and non-productivity of aleph_2-cc

Uniform n-place functions on T\subseteq ds(α)

In this paper the Erdos-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class rs(infty)^{< aleph_0} (finite sequences of decreasing sequences of ordinals) with aleph_0 equivalence classes, and for n< omega a notion of n-end-uniformity for a colouring of rs(infty)^{< aleph_0} with mu colours. We then show that for every ordinal alpha, n< omega and cardinal mu there is an ordinal lambda so that for any colouring c of T=rs(lambda)^{< aleph_0} with mu colours, T contains S isomorphic to rs(alpha) so that c rest S^{< aleph_0} is n-end uniform…

On the number of L_{infty,omega_1}-equivalent non-isomorphic models

We prove that if ZF is consistent then ZFC+GCH is consistent with the following statement: There is for every k<omega a model of cardinality aleph_1 which is L_{infty,omega_1}-equivalent to exactly k non-isomorphic models of cardinality aleph_1. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC+GCH that there is a ``good'' ladder system having exactly p^m pairwise nonequivalent colourings.

不可数序数上的行走与高阶Aronszajn线的非结构定理

Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

结合序数上的行走、club guessing和强着色等组合方法，证明了若存在$\aleph_2$-Aronszajn线则存在不含$\aleph_2$-Countryman线的特例，解决了Moore的问题，并将相关非结构定理推广至多种基数情形。