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]<sup>ω</sup>, providing a description of how far away a front is from being a barrier.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:679191 |
| Date | January 2015 |
| Creators | Mckay, Gregory |
| Publisher | University of East Anglia |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ueaeprints.uea.ac.uk/56893/ |
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