Related papers: Ramsey Classes: Examples and Constructions
We derive a new sufficient condition for the existence of {\omega}-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubi\v{c}ka and Ne\v{s}et\v{r}il.…
We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it…
The theory of two binary relations has the strong amalgamation property when the first relation is assumed to be coarser than the second relation, and each relation satisfies a chosen set of properties from the following list: transitivity,…
We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized)…
We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction.
We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple…
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…
We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todor\v{c}evi\'c correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of…
Generalised indiscernibles highlight a strong link between model theory and structural Ramsey theory. In this paper, we use generalised indiscernibles as tools to prove results in both these areas. More precisely, we first show that a…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…
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…
By a result of Zucker, every Fra\"iss\'e structure $\bf F$ for which the elements of $\mathrm{Age}(\bf F)$ have finite Ramsey degrees admits a Fra\"iss\'e precompact expansion $\bf F^{*}$ whose age $\mathrm{Age}(\bf F^{*})$ has the Ramsey…
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…
Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of…
We give a parametrization with perfect subsets of $2^{\infty}$ of the abstract Ramsey theorem (see \cite{todo}) Our main tool is an extension of the parametrized version of the combinatorial forcing developed in \cite{nash} and \cite{todo},…
We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures…
Motivated by Tukey classification problems and building on work in \cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramsey spaces $\mathcal{R}_{\alpha}$, $\alpha<\omega_1$. These spaces form a natural hierarchy of…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
Recently, Solecki introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman's theorem, Carlson's theorem, and Gowers' FIN$_k$ theorem. He proved that an entire class of finite monoids is Ramsey.…
A class of structures is said to have the homomorphism-preservation property just in case every first-order formula that is preserved by homomorphisms on this class is equivalent to an existential-positive formula. It is known by a result…