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Proving a witness lemma in better-quasiordering theory: the method of ‘extensions’

Published online by Cambridge University Press:  28 June 2011

Igor Kříž
Igor Kříž
Affiliation:
The Ohio State University, Department of Mathematics, Columbus, Ohio 43210, U.S.A.
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Abstract

We introduce a new method for proving ‘witness lemmas’ in BQO theory. As an application, we obtain witness-lemmas for arbitrary trees and for σ-scattered linear orderings. A positive answer to a question by Van Engelen, Miller and Steel follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 253 - 262
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Engelen, F. Van, Miller, A. W. and Steel, J.. Rigid Borel sets and better quasiorder theory. Contemp. Math. 65 (1987), 119–222.Google Scholar
[2] Friedman, H., Robertson, N. and Seymour, P. D.. The metamathematics of the graph minor theorem. Contemp. Math. 65 (1987), 229–261.Google Scholar
[3] Galvin, F. and Prikry, K.. Borel sets and Ramsey's theorem. J. Symbolic Logic 38 (1973), 193–198.Google Scholar
[4] Higman, G.. Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2 (1952), 326–336.CrossRefGoogle Scholar
[5] Kruskal, J.. Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Trans. Amer. Math. Soc. 95 (1960), 210–225.Google Scholar
[6] Laver, R.. On Fraisé's order type conjecture. Ann. of Math. 93 (1971), 89–111.CrossRefGoogle Scholar
[7] Laver, R.. Better-quasi-ordering and a class of trees. In Studies in Foundations and Combinatorics, Advances in Math. Suppl. Stud. no. 1 (Academic Press, 1978), pp. 31–48.Google Scholar
[8] Mansfield, R. and Weitkamp, G.. Recursive Aspects of Descriptive Set Theory. Oxford Logic Guides no. 11 (Oxford University Press, 1985).Google Scholar
[9] Nash-Williams, C. St. J. A.. On well-quasi-ordering finite trees. Proc. Cambridge Philos. Soc. 59 (1963), 833–835.Google Scholar
[10] Nash-Williams, C. St. J. A.. On better-quasi-ordering transfinite sequences. Proc. Cambridge Philos. Soc. 64 (1968), 273–290.Google Scholar
[11] Nash-Williams, C. St. J. A.. On well-quasi-ordering infinite trees. Proc. Cambridge Philos. Soc. 61 (1965), 697–720.Google Scholar
[12] Simpson, S.. BQO Theory and Fraisé's Conjecture. Chapter 9 of [8].Google Scholar
[13] Thomas, R.. A counter-example to ‘Wagner's conjecture’ for infinite graphs. Math. Proc. Cambridge Philos. Soc. 103 (1988), 55–57.Google Scholar
[14] Thomas, R.. Well-quasi-ordering infinite graphs with forbidden finite planar minor. (Submitted for publication.)Google Scholar