Related papers: Ramsey-like theorems for separable permutations
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.…
A graph $G$ is semilinear of complexity $t$ if the vertices of $G$ are elements of $\mathbb{R}^{d}$ for some $d\in\mathbb{Z}^{+}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions…
One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided…
The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a…
An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…
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…
One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the self-embedding monoid, the…
Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there…
We show that the Ramsey number is linear for every uniform hypergraph with bounded-degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chv\'atal et al. showed in 1983. Our proof is simple, contains the…
A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the…
We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for…
We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show…
We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory…
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest…
The rainbow Ramsey theorem states that every coloring of tuples where each color is used a bounded number of times has an infinite subdomain on which no color appears twice. The restriction of the statement to colorings over pairs (RRT22)…
Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied…
We study the uniform computational content of Ramsey's theorem in the Weihrauch lattice. Our central results provide information on how Ramsey's theorem behaves under product, parallelization and jumps. From these results we can derive a…
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…