Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function is nondecreasing on some set in U. Natural weakenings of these properties led to the inequivalent notions of weakly Ramsey and of quasi-selective ultrafilter, introduced and studied in [1] and [4], respectively. U is weakly Ramsey if for every finite colouring of unordered pairs of integers there is a set in U whose pairs share only two colours, while U is f-quasi-selective if every function g < f is nondecreasing on some set in U. (So the quasi-selective ultrafilters of [4] are here id-quasi selective.) In this paper we consider the relations between various natural cuts of the ultrapowers of N modulo weakly Ramsey and f-quasi-selective ultrafilters. In particular we characterize those weakly Ramsey ultrafilters that are isomorphic to a quasi-selective ultrafilter.
We investigate families of partitions of omega which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on the set of partition-filters and consider the dual form of some cardina…
We give a characterizations of Ramsey ultrafilters on $\mathscr P(ω)$ in terms of functions $f:ω^n\toω$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[ω]^n$ there is a finite partition~$\mathcal Q$ of $[ω]^{2n}$ such that any set $X\subseteq ω$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.
Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper developes a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to …
Associated to each ultrafilter $\mathcal{U}$ on $ω$ and each map $p:ω\rightarrow ω$ is a Dedekind cut in the ultrapower $ω^ω/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space $\mathcal{R}_{1}$. Associated to the space $\mathcal{R}_{1}$ is a notion of Ramsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion of Ramsey ultrafilter on $ω$. We characterize, under CH, the cuts …
We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erd{ö}s and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various specia…
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 the relation of "almost-reduction" in an arbitrary topological Ramsey space R, as a generalization of the relation of "almost-inclusion" on the space of infinite sets of natural numbers (the Ellentuck space). This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of "selective ultrafilter" on N, the set of natural numbers. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space to the u…
Posgrado Conjunto en Ciencias Matemáticas UNAM-UMSNH（UNAM-UMSNH数学科学联合研究生项目）；Escuela Superior de Fśica y Matemáticas Instituto Politécnico Nacional（国立理工学院高等物理与数学学院）；Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Campus Morelia（墨西哥国立自治大学莫雷利亚校区数学科学中心）
Motivated by a Tukey classification problem we develop here a new topological Ramsey space $\mathcal{R}_1$ that in its complexity comes immediately after the classical is a natural Ellentuck space \cite{MR0349393}. Associated with $\mathcal{R}_1$ is an ultrafilter $\mathcal{U}_1$ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on $\mathcal{R}_1$. This is analogous to the Pudlak-\Rodl\ Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keis…
Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this article is to introduce ultrafilters in a friendly manner and present some applications to the branch of combinatorics known as Ramsey theory, culminating with a new ultrafilter-based proof of van der Waerden's theorem.
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space $\mathcal{R}$ has an associated notion of Ramsey ultrafilter for $\mathcal{R}$ and selective ultrafilter for $\mathcal{R}$ (see \cite{MijaresSelective}). If $\mathcal{R}$ is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on $ω$; so by a well-known result of Kunen the two are equivalent. We…
In standard construction of hyperrational numbers using an ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with selective ultrafilter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only $α$-many iterations for some countable ordinal $α$. Here we study such $α$-iterable cardinals. We show that the $α$-iterable cardinals form a strict hierarchy for $α\leqω_1$, that they are downward absolute to $L$ for $α<ω_1^L…
We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of "asymptotic numerosities" for all sets of tuples of natural numbers. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptot…
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…
Motivated by Tukey classification problems and building on work in \cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramsey spaces $\mathcal{R}_α$, $α<ω_1$. These spaces form a natural hierarchy of complexity, $\mathcal{R}_0$ being the Ellentuck space, and for each $α<ω_1$, $\mathcal{R}_{α+1}$ coming immediately after $\mathcal{R}_α$ in complexity. Associated with each $\mathcal{R}_α$ is an ultrafilter $\mathcal{U}_α$, which is Ramsey for $\mathcal{R}_α$, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems…
We study the definability of ultrafilter bases on $ω$ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in $L$ we can construct $Π^1_1$ P-point and Q-point bases. We also show that the existence of a $\mathbfΔ^1_{n+1}$ ultrafilter is equivalent to that of a $\mathbfΠ^1_n$ ultrafilter base, for $n \in ω$. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.
The generic ultrafilter $\mathcal{G}_2$ forced by $\mathcal{P}(ω\timesω)/($Fin$\otimes$Fin) was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters (in a recent paper of Blass, Dobrinen, and Raghavan), but it was left open where exactly in the Tukey order it lies. We prove that $\mathcal{G}_2$ is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each $k\ge 2$, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter $\mathcal{G}_k$ forced by $\mathcal{P}(ω^k)/$Fin$^{\otimes k}$ forms a …
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 study ultrafilters on $ω^2$ produced by forcing with the quotient of $\scr P(ω^2)$ by the Fubini square of the Fréchet filter on $ω$. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at …
We study the partition properties enjoyed by the "next best thing to a P-point'' ultrafilters introduced recently in joint work with Dobrinen and Raghavan. That work established some finite-exponent partition relations, and we now analyze the connections between these relations for different exponents and the notion of conservativity introduced much earlier by Phillips. In addition, we establish some infinite-exponent partition relations for these ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed by selective 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.
We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN.
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…
A set X which is a subset of the Cantor set has property (s) (Marczewski (Spzilrajn)) iff for every perfect set P there exists a perfect set Q contained in P such that Q is a subset of X or Q is disjoint from X. Suppose U is a nonprincipal ultrafilter on omega. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answ…
It is shown that the known notion of selective coideal can be extended to a family $\mathcal{H}$ of subsets of $\mathcal{R}$, where $(\mathcal{R},\leq,r)$ is a topological Ramsey space in the sense of Todorcevic (see \cite{todo}). Then it is proven that, if $\mathcal{H}$ selective, the $\mathcal{H}$-Ramsey and $\mathcal{H}$-Baire subsets of $\mathcal{R}$ are equivalent. This extends the results of Farah in \cite{farah} for semiselective coideals of $\mathbb{N}$. Also, it is proven that the family of ${\cal H}$--Ramsey subsets of ${\cal R}$ is closed under the Souslin operation.
Given a topological Ramsey space $(\mathcal R,\leq, r)$, we extend the notion of semiselective coideal to sets $\mathcal H\subseteq\mathcal R$ and study conditions for $\mathcal H$ that will enable us to make the structure $(\mathcal R,\mathcal H,\leq, r)$ a Ramsey space (not necessarily topological) and also study forcing notions related to $\mathcal H$ which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results of Farah, and results of Mijares, to the most general context of topological Ramsey spa…
CONTENTS: On Selective screenability and examples of R. Pol. Workshops and conferences: The Oxford Conference on Topology and Computer Science in Honour of Peter Collins and Mike Reed; Boise Extravaganza In Set Theory (BEST2006). Research announcements: The isometry group of the Urysohn space as a Levy group; Chasing Silver; Disjoint Non-Free Subgoups of Abelian Groups; Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts; A Vitali set can be homeomorphic to its complement; o-Boundedness of free objects over a Tychonoff space; On the consistency strength of the Mil…
