Thursday, February 10, 2022 at 11:00 a.m.
PhD Candidate: Keegan Dasilva Barbosa
Supervisor: Stevo Todorcevic
Thesis title: Ramsey Degree Theory of Ordered and Directed Sets
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We will study various Ramsey degree problems pertaining to categories of structures under various
embedding types. Using a technique originally penned by Laver, we show that the class of Aronszajn
lines ordered under the embedding relation is a better quasi-order when we assume PFA. As a corollary,
we deduce one dimensional Ramsey degrees for Aronszajn lines. We also devise a colouring algorithm to
colour Borel graphs coded by better quasi-orders on countable sets. We apply our algorithm to various
types of well established better quasi-orders and deduce that there is a whole class of better quasi-orders
that exists without a single known constructible example. We will conclude with some results pertaining
to the Kechris-Pestov-Todorcevic correspondence. This will include a categorical notion of precompact
expansion, which will prove to be more versatile in computing Ramsey degrees, as well as a weaker
notion of the Ramsey property which also corresponds to xed point properties of automorphism groups
of ultrahomogeneous structures. This includes an application to trees under various embedding types,
including the foundational strong embedding types studied by Milliken.
A copy of the thesis can be found here: Keegan Dasilva Barbosa Thesis
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