arXiv subjects
Publications and source records attributed to Keegan Dasilva Barbosa.
We extend the Kechris--Pestov--Todorčević correspondence to weak Fra\"ıssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders, and categories of strong embeddings of trees.
Every statement of the Ramsey theory of finite structures corresponds to the fact that a particular category has the Ramsey property. We can, then, compare the strength of Ramsey statements by comparing the ``Ramsey strength'' of the corresponding categories. The main thesis of this paper is that establishing pre-adjunctions between pairs of categories is an appropriate way of comparing their ``Ramsey strength''. What comes as a pleasant surprise is that pre-adjunctions generalize the Tukey reducibility in the same way categories generalize preorders. In this paper we set forth a classification program of statements of finite Ramsey theory based on their relationship with respect to this generalized notion of Tukey reducibility for categories. After identifying the ``weakest'' Ramsey category, we prove that the Finite Dual Ramsey Theorem is as powerful as the full-blown version of the Graham-Rothschild Theorem, and conclude the paper with the hypothesis that the Finite Dual Ramsey Theorem is the ``strongest'' of all finite Ramsey statements.
In this short note we confirm the deep structural correspondence between the complexity of a countable scattered chain (= strict linear order) and its big Ramsey combinatorics: we show that a countable scattered chain has finite big Ramsey degrees if and only if it is of finite Hausdorff rank. This also provides a complete characterization of countable chains whose big Ramsey spectra are finite. We expand the notion of big Ramsey spectrum to monomorphic structures and give a sufficient condition for a monomorphic countable structure to have finite big Ramsey spectrum.
This paper introduces the concept of a productive notion of big Ramsey degree and showcases its versatility through a handful of applications. The main focus is notably providing sufficient conditions for the existence of a finite canonical basis of equivalence relations, building upon the prior work of Laflamme, Sauer, and Vuksanovic. Additionally, a combinatorial analysis of indexed structures is conducted.
Every better quasi-order codifies a Borel graph that does not contain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic number, though one has yet to be explicitly constructed. In this paper, we show that examples cannot be constructed via standard methods. Moreover, we show that most of the known better quasi-orders are non-examples, suggesting there is still a class of better quasi-orders with interesting combinatorial properties who's elements/members still remain unknown.
We show that under the proper forcing axiom the class of all Aronszajn lines behave like $σ$-scattered orders under the embeddability relation. In particular, we are able to show that the class of better quasi order labeled fragmented Aronszajn lines is itself a better quasi order. Moreover, we show that every better quasi order labeled Aronszajn line can be expressed as a finite sum of labeled types which are algebraically indecomposable. By encoding lines with finite labeled trees, we are also able to deduce a decomposition result, that for every Aronszajn line $L$ there is integer n such that for any finite colouring of $L$ there is subset $L^\prime$ of $L$ isomorphic to $L$ which uses no more than n colours.
We generalize the notion of relational precompact expansions of Fraïssé classes via functorial means, inspired by the technique outlined by Laflamme, Nguyen Van Thé and Sauer in their paper Partition properties of the dense local order and a colored version of Milliken's theorem arXiv:0710.2885. We also generalize the expansion property and prove that categorical precompact expansions grant upper bounds for Ramsey degrees. Moreover, we show under strict conditions, we can also compute big Ramsey degrees. We also apply our methodology to calculate the big and little Ramsey degrees of the objects in Age$(\mathbf{S}(n))$ for all $n\geq 2$.