arXiv subjects

Nadav Meir

Publications and source records attributed to Nadav Meir.

8 recordsLinked to original sources

arXiv2024

Generalised Indiscernibles, Dividing Lines, and Products of Structures

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 reduct of an ultrahomogenous $\aleph_0$-categorical structure which has higher arity than the original structure cannot be Ramsey. In particular, the only nontrivial Ramsey reduct of the generically ordered random $k$-hypergraph is the linear order. We then turn our attention to model-theoretic dividing lines that are characterised by collapsing generalised indiscernibles, and prove, for these dividing lines, several transfer principles in (full and lexicographic) products of structures. As an application, we construct new algorithmically tame classes of graphs.

math.LO↗

arXiv2023

The externally definable Ramsey property and fixed points on type spaces

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for ultrahomogeneous structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for each natural number n, every subflow of the space of n-types with parameters in M has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.

math.LO↗

arXiv2023

All These Approximate Ramsey Properties

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 structures in which every structure of size at most 2 is a "Ramsey object" essentially consists of ordered structures, generalising a known result for countable Ramsey classes.

math.CO↗

arXiv2023

Practical and Structural Infinitary Expansions

Given a structure $M$ we introduce infinitary logic expansions, which generalise the Morleyisation. We show that these expansions are tame, in the sense that they preserve and reflect both the Embedding Ramsey Property (ERP) and the Modelling Property (MP). We then turn our attention to Scow's theorem connecting generalised indiscernibles with Ramsey classes and show that by passing through infinitary logic, one can obtain a stronger result, which does not require any technical assumptions. We also show that every structure with ERP, not necessarily countable, admits a linear order which is a union of quantifier-free types, effectively proving that any Ramsey structure is ``essentially'' ordered. We also introduce a version of ERP for classes of structures which are not necessarily finite (the finitary-ERP) and prove a strengthening of the Kechris-Pestov-Todorcevic correspondence for this notion.

math.LO↗

arXiv2019

Many symmetrically indivisible structures

A structure $\mathcal{M}$ in a first-order language $\mathcal{L}$ is \emph{indivisible} if for every coloring of $M$ in two colors, there is a monochromatic $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ such that $\mathcal{M}^{\prime}\cong\mathcal{M}$. Additionally, we say that $\mathcal{M}$ is symmetrically indivisible if $\mathcal{M}^{\prime}$ can be chosen to be \emph{symmetrically embedded} in $\mathcal{M}$ (that is, every automorphism of $\mathcal{M}^{\prime}$ can be extended to an automorphism of $\mathcal{M}$). In the following paper we give a general method for constructing new symmetrically indivisible structures out of existing ones. Using this method, we construct $2^{\aleph_0}$ many non-isomorphic symmetrically indivisible countable structures in given (elementary) classes and answer negatively the following question asked by A. Hasson, M. Kojman and A. Onshuus in "On symmetric indivisibility of countable structures" (Cont. Math. 558(1):453--466): Let $\mathcal{M}$ be a symmetrically indivisible structure in a language $\mathcal{L}$. Let $\mathcal{L}_0 \subseteq \mathcal{L}$. Is $ \mathcal{M} \upharpoonright \mathcal{L}_0$ symmetrically indivisible?

math.LO↗

arXiv2019

On products of elementarily indivisible structures

We say a structure $M$ in a first-order language is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure $M'$ of $M$ such that $M'$ is isomorphic to $M$. Additionally, we say that $M$ is symmetrically indivisible if $M'$ can be chosen to be symmetrically embedded in $M$ (that is, every automorphism of $M'$ can be extended to an automorphism of $M$). Similarly, we say that $M$ is elementarily indivisible if $M'$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman and A. Onshuus.

math.LO↗

arXiv2019

Pseudo-finite sets, pseudo-o-minimality

We give an example of two ordered structures M, N in the same language L with the same universe, the same order and admitting the same one-variable definable subsets such that M is a model of the common theory of o-minimal L-structures and N admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two questions by Schoutens; the first being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle.

math.LO↗

arXiv2019

Infinite Lexicographic Products

We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.

math.LO↗