Related papers: Finite Ramsey Theory through Category Theory
We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions…
This list presents problems in the Reverse Mathematics of infinitary Ramsey theory which I find interesting but do not personally have the techniques to solve. The intent is to enlist the help of those working in Reverse Mathematics to take…
Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a…
We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making…
This paper contains a stronger version of a final identification theorem for the `generic' groups of finite Morley rank.
We classify a sharp phase transition threshold for Friedman's finite adjacent Ramsey theorem. We extend the method for showing this result to two previously known classifications involving Ramsey theorem variants: the Paris--Harrington…
Extending a result of K. Milliken \cite{Mi2}, in this paper we prove a Ramsey classification result for equivalence relations defined on uniform families of finite strong subtrees of a finite sequence $(U_i)_{i\in d}$ of fixed trees $U_i$,…
The purpose of these notes is to collect in one place some facts on the category of finite totally ordered sets and some related categories. More specifically, we collect some results on them which will be useful for the study of iteratedly…
A new definition for the notion of a (general) $\infty$-category is given.
Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on…
We prove a new characterization of the Ramsey property of categories in terms of a generalized form of K\H{o}nig's tree lemma. Afterwards, we discuss its applications to structural Ramsey theory. In particular, we provide a new proof of the…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…
In this paper we show that a countable structure admitting a finite monomorphic decomposition has finite big Ramsey degrees if and only if so does every monomorphic part in its minimal monomorphic decomposition. The necessary prerequisite…
As the first part of the treatise on A General Theory of Concept Lattice (I-V), this work develops the general concept lattice for the problem concerning categorization of objects according to their properties. Unlike the conventional…
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…
The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal,…
We restructure and advance the classification theory of finite racks and quandles by employing powerful methods from transformation groups and representation theory, especially Burnside rings. These rings serve as universal receptacles for…
We work with $FI$-modules over a small preadditive category $\mathcal R$, viewed as a ring with several objects. Our aim is to study torsion theories for $FI$-modules. We are especially interested in torsion theories on finitely generated…
Using the Carlson-Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees
This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to…