Canonical models for aleph_1 combinatorics

A note on canonical functions

We construct a generic extension in which the aleph_2 nd canonical function on aleph_1 exists.

Note on Representing $\aleph_0$-categorical Linear Orders

These notes find a canonical representation of the $\aleph_0$-categorical linear orders based on Joseph Rosenstein's description. A unique minimal representation, called the normal form, is obtained.

任意共尾性的第一奇异基数后继处可逼近性的失效

Failure of Approachability at the Successor of the first Singular for any Cofinality

该研究解决了两个关于ℵ_{ω+1}组合学的长期公开问题，证明对任意n≥1，GCH成立且存在共尾性ℵ_n的驻集不在可逼近性理想中，并得到良态性与可逼近性在共尾性ℵ_1的驻点上不同的模型。

On the canonical base property

We give an example of a finite rank, in fact aleph-1 categorical theory where the CBP (canonical base property) does not hold. We include a "group-like" example. We also prove, in a finite Morley rank context, if all definable Galois groups are "rigid" then the theory has the CBP (in a strong sense).

An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowski's Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first author's result th…

On the universality of the nonstationary ideal

Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal…