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Title: Better-quasi-orders : extensions and abstractions
Author: Mckay, Gregory
ISNI:       0000 0004 5371 4226
Awarding Body: University of East Anglia
Current Institution: University of East Anglia
Date of Award: 2015
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We generalise the notion of δ-scattered to partial orders and prove that some large classes of δ-scattered partial orders are better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomassé regarding δ-scattered linear orders, δ-scattered trees, countable pseudo-trees and N-free partial orders. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order. We prove that some natural classes of structured δ-scattered pseudo-trees are better quasi-ordered, strengthening similar results of Kříž, Corominas and Laver. We then use this theorem to prove that some large classes of graphs are better-quasi-ordered under the induced subgraph relation, thus generalising results of Damaschke and Thomassé. We investigate abstract better-quasi-orders by modifying the normal definition of better-quasi-order to use an alternative Ramsey space rather than exclusively the Ellentuck space as is usual. We classify the possible notions of well-quasi-order that can arise by generalising in this way, before proving that the corresponding notion of better-quasi-order is closed under taking iterated power sets, as happens in the usual case. We consider Shelah's notion of better-quasi-orders for uncountable cardinals, and prove that the corresponding modification of his definition using fronts instead of barriers is equivalent. This gives rise to a natural version of Simpson's definition of better-quasi-order for uncountable cardinals, even in the absence of any Ramsey-theoretic results. We give a classification of the fronts on [K]ω, providing a description of how far away a front is from being a barrier.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available