Related papers: Abstract approach to finite Ramsey theory and a se…
We define a collection of topological Ramsey spaces consisting of equivalence relations on $\omega$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $\omega$. To prove…
It has become obvious in the recent development that the structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this paper we explicitely put the…
We survey recent developments concerning two properties of classes of finite structures: the Ramsey property and the extension property for partial automorphisms (EPPA).
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories of indiscernibles" to be special…
In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup deformations of semigroups. In this context, new notions of Ramsey principle for…
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the author's recent method of trees with coding nodes are…
The main results of this paper (a) extend the finite Ramsey partition theorem, and (b) employ this extension to obtain a stronger form of the infinite Nash-Williams partition theorem, and also a new proof of Ellentuck's, and hence…
We prove a generalization of the infinite quantum Ramsey theorem of Kennedy et al. (arXiv:1711.09526), showing that it follows from an archetypical "selective" pattern satisfied by certain families of projections in an infinite-dimensional…
Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed…
This short note establishes an abstract Hales--Jewett theorem for semigroups equipped with a finite family of retractions. The proof relies on the interplay between retractions and tensor products of ultrafilters.
Application of the Ramsey Infinite Theorem to the variational principles of physics is discussed. According to the Ramsey Infinite Theorem,there exists the infinite, monochromatic chain of the pathways (clique), which are completely built…
We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected…
The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the…
In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Ne\v{s}et\v{r}il and V. R\"{o}dl claims that the class of all finite posets…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
Ramsey's theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this…
The celebrated theorem of Kechris, Pestov and Todor\v{c}evi\'c connecting structural Ramsey theory with topological dynamics has as a consequence that the Fra\"{\i}ss\'e limit of a Ramsey class of non-trivial finite relational structures…
In 2012 M. Soki\'c proved that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. B\"ottcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another…
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic…
We consider finitary approximations of the (embedding) Ramsey property. Using a class of homogeneous reducts of random ordered hypergraphs, we prove that these properties form a strict hierarchy. We also show that every class of finite…