Related papers: Comparisons of polychromatic and monochromatic Ram…
We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{\omega_2}]^2\to 2$ establishing $\omega_2 \not\to [(\omega_1;{\omega})]^2_2$ such that some colouring…
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…
A major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemer\'edi's theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs…
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…
The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for…
The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…
A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…
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…
Hindman's theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman's Theorem. This paper introduces Ramsey algebras and…
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…
Given a metric space $(X,d)$, we say that a mapping $\chi: [X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$ implies $\chi(\{x,y\})=\chi(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an infinite metric space…
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…
The main result of this paper is a "colored Tverberg theorem for rainbow-unavoidable complexes". This theorem may be considered as a merging of two theorems: "Tverberg theorem for collectively unavoidable complexes" and "balanced colored…
The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the…
The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic…
When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and…
We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if $\ell_m$ denotes $m$ collinear points with consecutive points of distance one apart, we…
An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the…
The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's…
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…