Better-quasi-orders : extensions and abstractions

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.