Jindrich Zapletal
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
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Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on forcings which contain dense subsets forming topological Ramsey spaces. These forcings motivated the development of new Ramsey theory, which then was applied to the generic ultrafilters to obtain the precise structure Rudin-Keisler and Tukey orders below such ultrafilters. The content of the first two tutorials has appeared in an expository article submitted to th…
Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a $σ$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[ω]^ω$. Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler a…
This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples.
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[ω_1]^2\rightarrowω$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of posi…
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 …
We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $ω$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$.
We define the dualizations of objects and concepts which are essential for investigating the Ramsey property in the first levels of the projective hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and dual Laver forcing, and show that the Harrington-Kechris techniques for proving the Ramsey property from determinacy work in the dualized case as well.
In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet). Further we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries…
We show that the set of codes for Ramsey positive analytic sets is $\mathbfΣ^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is $\mathbfΣ^1_1$-complete. This shows a close resemblance between the Sacks forcing and the Mathias forcing. In particular, we get that the $σ$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem posed by Ikegami, Pawlikowski and Zapletal.
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…
We show that, assuming GCH, if $κ$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $κ$ and obeying the constraints of Easton's theorem, namely, $F(α)\leq F(β)$ for $α\leqβ$ and $α<\cf(F(α))$, then there is a cofinality preserving forcing extension in which $κ$ remains Ramsey or strongly Ramsey respectively and $2^δ=F(δ)$ for every regular cardinal $δ$.
Following the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok and J. Zapletal we study Borel equivalences on Laver trees. Here we prove that equivalence relations Borel reducible to an equivalence relation on $2^ω$ given by some $F_σ$ $P$-ideal on $ω$ can be canonized to the full equivalence relation or to the identity relation. This has several corollaries, e.g. Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by $F_σ$ $P$-ideals.
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…
With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S …
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.
Ramsey theorem for trees with successor operation
本文证明带后继运算的树的一般 Ramsey 定理，推广 Carlson-Simpson 与 Milliken 树定理，并应用于 Nešetřil-Rödl 和 Graham-Rothschild 定理，及大 Ramsey 度的非迫力证明。
We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014. Given a set $X$, let $[X]^n$ be the collection of all $n$-element subsets of $X$. Ramsey's theorem for $n$-tuples asserts the existence, for every finite coloring of $[ω]^n$, of an infinite set $X \subseteq ω$ such that $[X]^n$ is monochromatic. The meta-mathematical study of Ramsey has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's t…
We show in this note that in the forcing extension by $Add(ω,\beth_ω)$, the following Ramsey property holds: for any $r\in ω$ and any $f: \mathbb{R}\to r$, there exists an infinite $X\subset \mathbb{R}$ such that $X+X$ is monochromatic under $f$. We also show the Ramsey statement above is true in $\mathrm{ZFC}$ when $r=2$. This answers two questions by Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky.
A Ramsey theorem for the reals
本文在 ZFC 中证明：对实数对的任意有限着色，存在与有理数同胚的集合至多用两种颜色，并推广至一大类拓扑空间。
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeabili…
Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic
本文在BΣ^0_3内构建了RT^2的\ll^2-解，并证明了其基理定理，同时展示了RT^2_2和EM_{<\infty}的类似定理，为相关保守性结果提供了更简单的证明。
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