arXiv subjects

Lynn Scow

Publications and source records attributed to Lynn Scow.

9 recordsLinked to original sources

arXiv2022

Big Ramsey Degrees in Ultraproducts of Finite Structures

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) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $\fLL^*$ has, as a spine, $\eta_1$, an uncountable analogue of the order type of rationals $\eta$. Finite big Ramsey degrees for $\eta$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $\eta_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $\eta_1$ to show that it witnesses big Ramsey degrees of finite tuples in $\eta$ on every copy of $\eta$ in $\eta_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

math.LO↗

arXiv2022

A new perspective on semi-retractions and the Ramsey property

We investigate the notion of a semi-retraction between two first order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We further these connections between combinatorics and model theory, and look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

math.LO↗

arXiv2021

Products of Classes of Finite Structures

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this paper do not yield new classes of theories.

math.LO↗

arXiv2020

Ramsey transfer to semi-retractions

We introduce the notion of a {\it semi-retraction}. Given two structures $\A$ and $\B$, $\A$ is a semi-retraction of $\B$ if there exist quantifier-free type respecting maps $f: \B \raw \A$ and $g: \A \raw \B$ such that $f \circ g$ is an embedding. We say that a structure has the Ramsey property if its age does. Given two locally finite ordered structures $\A$ and $\B$, if $\A$ is a semi-retraction of $\B$ and $\B$ has the Ramsey property, then $\A$ also has the Ramsey property. We introduce notation for what we call semi-direct product structures, after the group construction known to preserve the Ramsey property.~\cite{kpt05} We introduce the notion of a color-homogenizing map, and use this notion to give a finitary argument that the semi-direct product structure of ordered relational structures with the Ramsey property must also have the Ramsey property. Finally, we characterize NIP theories using a generalized indiscernible sequence indexed by a semi-direct product structure.

math.LO↗

arXiv2015

Categorical Constructions and the Ramsey Property

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 Ramsey property and the dual Ramsey property in the context of categories of finite structures and investigate the invariance of these properties under some standard categorical constructions. We use elementary category theory to generalize some combinatorial results and using the machinery of very basic category theory provide new combinatorial statements (whose formulations do not refer to category-theoretic notions).

math.CT↗

arXiv2015

Characterizing Model-Theoretic Dividing Lines via Collapse of Generalized Indiscernibles

We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.

math.LO↗

arXiv2014

Indiscernibles, EM-types, and Ramsey Classes of Trees

It was shown in \cite{sc12} that for a certain class of structures $\I$, $\I$-indexed indiscernible sets have the modeling property just in case the age of $\I$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we may conclude that certain classes of finite trees are Ramsey, some previously known. See updated paper for new references.

math.LO↗

arXiv2013

Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set $\{a_η\mid η\in \W\}$ that saw original use in \cite{sh90}, which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.

math.LO↗

arXiv2011

Characterization of NIP theories by ordered graph-indiscernibles

We generalize the Unstable Formula Theorem characterization of stable theories from \citep{sh78}: that a theory $T$ is stable just in case any infinite indiscernible sequence in a model of $T$ is an indiscernible set. We use a generalized form of indiscernibles from \citep{sh78}: in our notation, a sequence of parameters from an $L$-structure $M$, $(b_i : i \in I)$, indexed by an $L'$-structure $I$ is \emph{$L'$-generalized indiscernible in $M$} if qftp$^{L'}(\ov{i};I)$=qftp$^{L'}(\ov{j};I)$ implies tp$^L(\ov{b}_{\ov{i}}; M)$ = tp$^L(\ov{b}_{\ov{j}};M)$ for all same-length, finite $\ov{i}, \ov{j}$ from $I$. Let $T_g$ be the theory of linearly ordered graphs (symmetric, with no loops) in the language with signature $L_g=\{<, R\}$. Let $\K_g$ be the class of all finite models of $T_g$. We show that a theory $T$ has NIP if and only if any $L_g$-generalized indiscernible in a model of $T$ indexed by an $L_g$-structure with age equal to $\K_g$ is an indiscernible sequence.

math.LO↗