Better-Quasi-Orders: Extensions and Abstractions

Mckay, Gregory (2015) Better-Quasi-Orders: Extensions and Abstractions. Doctoral thesis, University of East Anglia.

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We generalise the notion of �-scattered to partial orders and prove that some large classes
of �-scattered partial orders are better-quasi-ordered under embeddability. This generalises
theorems of Laver, Corominas and Thomass�e regarding �-scattered linear orders,
�-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 satis�es a natural strengthening of better-quasi-order.
We prove that some natural classes of structured �-scattered pseudo-trees are betterquasi-
ordered, strengthening similar results of K�r���z, 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 Thomass�e.
We investigate abstract better-quasi-orders by modifying the normal de�nition 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 modi�cation of his de�nition using fronts instead of barriers is
equivalent. This gives rise to a natural version of Simpson's de�nition of better-quasiorder
for uncountable cardinals, even in the absence of any Ramsey-theoretic results. We
give a classi�cation of the fronts on [�]!, providing a description of how far away a front
is from being a barrier.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Users 2259 not found.
Date Deposited: 29 Jan 2016 12:30
Last Modified: 29 Jan 2016 12:30

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