Related papers: Canonizing structural Ramsey theorems
The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $\mathcal{X}\subseteq [\omega]^{\omega}$, where $[\omega]^{\omega}$ is endowed with the metric topology, each infinite…
Often regarded as the study of how order emerges from randomness, Ramsey theory has played an important role in mathematics and computer science, giving rise to applications in numerous domains such as logic, parallel processing, and number…
We study four families of consequences of Ramsey's Theorem from the viewpoint of reverse mathematics. The first, which we call the Achromatic Ramsey Theorem, is from a partition relation introduced by Erd\H{o}s, Hajnal and Rado: $\omega \to…
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse…
Given a graph $G$ and a collection $\mathcal C$ of subsets of $\mathbb{R}^d$ indexed by the subsets of vertices of $G$, a constrained drawing of $G$ is a drawing, where each edge is drawn inside some set from $\mathcal C$, in such a way…
Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a…
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…
Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In…
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…
Ramsey algebras are algebras that induce Ramsey spaces, which are generalizations of the Ellentuck space and Milliken's space. Previous work suggests a possible local version of Ramsey algebras induced by infinite sequences. Hence, we…
The class of finite distributive lattices, as many other classes of structures in everyday use, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriatelly chosen linear orders the…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…
We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.
An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an…
Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property.…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…
We define the continuous modeling property for first-order structures and show that a first-order structure has the continuous modelling property if and only if its age has the embedding Ramsey property. We use generalized indiscernible…
In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M.\ Soki\'c that the free product of Ramsey classes is a Ramsey class and the result of M.\ Bodirsky, M.\Pinsker and T.\…
We both survey and extend a new technique from Lu Liu to prove separation theorems between products of Ramsey-type theorems over computable reducibility. We use this technique to show that Ramsey's theorem for $n$-tuples and three colors is…