Techniques for approaching the dual Ramsey property in the projective hierarchy

A hierarchy of Ramsey-like cardinals

We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy.

A refinement of the Ramsey hierarchy via indescribability

A subset $S$ of a cardinal $κ$ is Ramsey if for every function $f:[S]^{<ω}\to κ$ with $f(a)<\min a$ for all $a\in[S]^{<ω}$, there is a set $H\subseteq S$ of cardinality $κ$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<ω$. Baumgartner proved \cite{MR0384553} that if $κ$ is a Ramsey cardinal, then the collection of non-Ramsey subsets of $κ$ is a normal ideal on $κ$. Sharpe and Welch \cite{MR2817562}, and independently Bagaria \cite{MR3894041}, extended the notion of $Π^1_n$-indescribability where $n<ω$ to that of $Π^1_ξ$-indescribability where…

Infinite dimensional Ellentuck spaces and Ramsey-classification theorems

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers $B$ on $ω$ as the prototype structures, we construct a class of continuum many topological Ramsey spaces $\mathcal{E}_B$ which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projection. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces $\mathcal{E}_B$, extending the Pudlak-Rodl Theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterati…

介于 Ramsey 与可测基数之间

Between Ramsey and measurable cardinals

本文研究介于 $κ$-Ramsey 基数与可测基数之间的交织层级，通过 filter games 刻画 Bovykin 和 McKenzie 提出的可测性 baby 版本，并确定直至可测基数的宇宙理论。

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of $σ$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $\mathcal{U}$ for $n$-tuples, denoted $t(\mathcal{U},n)$, is the smallest number $t$ such that given any $l\ge 2$ and coloring $c:[ω]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $σ$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at …

Controlling iterated jumps of solutions to combinatorial problems

Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their it…