A proof of Shelah's recent partition theorem

An Extension of Shelah's Trichotomy Theorem

In Sh506, Shelah develops the theory of $\mathrm{pcf}_I(A)$ without the assumption that $|A|<\min (A)$, going so far as to get generators for every $λ\in\mathrm{pcf}_I(A)$ under some assumptions on $I$. Our main theorem is that we can also generalize Shelah's trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for $\mathrm{pcf}_I(A)$ which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.

A Streamlined Proof of $\mathfrak{p}=\mathfrak{t}$

We streamline Malliaris and Shelah's proof that $\mathfrak{p} = \mathfrak{t}$. In particular, we replace cofinality spectrum problems with models of $ZFC^-$, and we eliminate the use of peculiar cuts.

Large normal ideals concentrating on a fixed small cardinality

A property of a filter, a kind of large cardinal property, suffices for the proof in Liu Shelah [LiSh:484] and is proved consistent as required there. A natural property which looks better, not only is not obtained here, but is shown to be false. On earlier related theorems see Gitik Shelah [GiSh310].

Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…

A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits

We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \g…

Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $λ$, a strong coloring on $λ^+$ by $λ$ colors over a partition can be stretched to one with $λ^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds…

Club guessing for dummies

We give a direct, detailed and relatively short proof of Shelah's theorem on club guessing sequences on $S^{μ^+}_μ$ (for any regular, uncountable cardinal $μ$).

关于无挠阿贝尔群的Borel完备性及一个Slaman-Wehner度谱

On the Borel completeness of torsion-free abelian groups and a Slaman-Wehner degree spectrum

本文给出无挠阿贝尔群Borel完备性的简化证明，并构造一个无计算副本但可从所有非计算集计算的群，解决Downey-Goncharov问题。

Boolean algebras with prescribed topological densities

We give a simplified proof of a theorem of M. Rabus and S. Shelah claiming that for each cardinal mu there is a c.c.c Boolean algebra with topological density mu.

Categoricity and infinitary logics

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $λ$ such that whenever $M, N \in K$ have size at least $λ$, $M \le N$ if and only if $M \preceq_{L_{\infty, \text{LS} (K)^+}} N$. The importance of the claim lies in the following theorem, implicit in Shelah's work: $\mathbf{Theorem}$ Assume the claim. Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then the class of $λ$ such that: 1) $K$ is categorical in $λ$; 2) $K$ has…

A note on the Revised GCH

We give a proof of Theorem 2.10 from [8] that eliminates the use of Shelah's nice filters and associated rank functions, and instead uses only the well-foundedness of reduced products of ordinals modulo countably complete filters. This removes any need to make assumptions about the existence of large cardinals in core models.

Eberlein-Smulian compactness and Kolmogorov extension theorems; a model theoretic approach

This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to E…

An inner model proof of the strong partition property of δ^2_1

We prove the strong partition property for δ^2_1 using methods from inner model theory.

A partition theorem for a large dense linear order

Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes, such that whenever any of these classes C is colored with fewer than K many colors, there is a copy Q* of Q_K such that all m-tuples from Q* in C receive the same color. As a consequence we obtain that whenever we color the m-tuples of Q with fewer than K many colors, there is a copy of Q_K all m-tuples from which are colored in at most t_m^+ colors. In other …

Polish space partition principles and the Halpern-Läuchli theorem

The Halpern-Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern-Läuchli theorem, and the same forcing from Harrington's proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

Partition theorems from creatures and idempotent ultrafilters

We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.