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Towards better: a motivated introduction to better-quasi-orders. (English) Zbl 1430.06001

Summary: The well-quasi-orders (WQO) play an important role in various fields such as Computer Science, Logic or Graph Theory. Since the class of WQOs lacks closure under some important operations, the proof that a certain quasi-order is WQO consists often of proving it enjoys a stronger and more complicated property, namely that of being a better-quasi-order (BQO).
Several articles – notably [T. Forster, Theor. Comput. Sci. 309, No. 1–3, 111–123 (2003; Zbl 1081.06002); J. B. Kruskal, J. Comb. Theory, Ser. A 13, 297–305 (1972; Zbl 0244.06002); R. Laver, Ann. Math. (2) 93, 89–111 (1971; Zbl 0208.28905); R. Laver, Math. Proc. Camb. Philos. Soc. 79, 1–10 (1976; Zbl 0405.06001); E. C. Milner, NATO ASI Ser., Ser. C 147, 487–502 (1985; Zbl 0573.06002); S. G. Simpson, “Bqo theory and Fraïssé’s conjecture”, in: Recursive aspects of descriptive set theory. Oxford: Oxford University Press. 124–138 (1985)] – contain valuable introductory material to the theory of BQOs. However, a textbook entitled “Introduction to better-quasi-order theory” is yet to be written. Here is an attempt to give a motivated and self-contained introduction to the deep concept defined by Nash-Williams that we would expect to find in such a textbook.

MSC:

06A06 Partial orders, general
06A07 Combinatorics of partially ordered sets
05D10 Ramsey theory
03E75 Applications of set theory
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