相关论文: A note on the Canonical Ramsey Theorem and Ramsey …

Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite…

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…

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…

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $\omega$ up to homeomorphism.…

Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms…

There is a long history of studying Ramsey theory using the algebraic structure of the Stone-\v{C}ech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic…

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…

Extending a result of K. Milliken \cite{Mi2}, in this paper we prove a Ramsey classification result for equivalence relations defined on uniform families of finite strong subtrees of a finite sequence $(U_i)_{i\in d}$ of fixed trees $U_i$,…

Let $G$ be a group, $X$ be an infinite transitive $G$-space. A free ultrafilter $\UU$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\PP$ of $X$, either one cell of $\PP$ is a member of $\UU$, or there is a member of…

We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-$\omega_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\textsf{CH}$, there is a unique…

Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which…

We show that, consistently, there is an ultrafilter F on omega such that if N^l_n=(P^l_n cup Q^l_n,P^l_n,Q^l_n,R^l_n) (for l =1,2, n< omega), P^l_n cup Q^l_n subseteq omega, and prod_{n<omega} N^1_n/F and prod_{n< omega}N^2_n/F are…

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as…

We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an…

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…

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles…

We say that a subset $S$ of an infinite group $G$ is a Ramsey-product subset if, for any infinite subsets $X$, $Y$ of $G$, there exist $x \in X$ and $y\in Y$ such that $x y \in S$ and $ y x \in S$ . We show that the family $\varphi$ of all…

A union ultrafilter is an ultrafilter over the finite subsets of $\omega$ that has a base of sets of the form $\mathrm{FU}(X)$, where $X$ is an infinite pairwise disjoint family and $\mathrm{FU}(X)=\{\bigcup…

Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if $P$ is any polynomial with no constant term, then for every finite coloring of $\mathbb{N}$, there exists two infinite subsets $B,C$ of $\mathbb{N}$ such that…