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On well-quasi-ordering infinite trees

Published online by Cambridge University Press:  24 October 2008

C. St. J. A. Nash-Williams
C. St. J. A. Nash-Williams
Affiliation:
King's College, Aberdeen
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Abstract

Let A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and s is x1, …, xm and t is x2, x3, …, xn. Call a subset S of A a P-block if, for every infinite ascending sequence x1, x2, … of positive integers, there exists an m such that x1, …, xm belongs to S. A quasi-ordered set Q (i.e. a set on which a reflexive and transitive relation ≤ is defined) is better-quasi-ordered if, for every P-block S and every function f:S → Q, there exist s, t ∈ S such that s ⊲ t and f(s) ≤ f(t). It is proved that any set of (finite or infinite) trees is better-quasi-ordered if T1 ≤ T2 means that the tree T1 is homeomorphic to a subtree of the tree T2. This establishes a conjecture of J. B.Kruskal that, if T1, T2, … is an infinite sequence of trees, then there exist i, j such that i < j and Ti ≤ Tj.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 697 - 720
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

(1)Higman, G.Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3) 2 (1952), 326–336.CrossRefGoogle Scholar
(2)Kruskal, J. B.Well-quasi-ordering, the tree theorem, and Vázsonyi's conjecture. Trans. Amer. Math. Soc. 95 (1960), 210–225.Google Scholar
(3)Nash-Williams, C. St.J. A.On well-quasi-ordering finite trees. Proc. Cambridge Philos. Soc. 59 (1963), 833–835.CrossRefGoogle Scholar
(4)Nash-Williams, C. St. J. A.On well-quasi-ordering lower sets of finite trees. Proc. Cambridge Philos. Soc. 60 (1964), 369–384.CrossRefGoogle Scholar
(5)Nash-Williams, C. St. J. A.On well-quasi-ordering tranafinite sequences. Proc. Cambridge Philos. Soc. (to appear).Google Scholar
(6)Rado, R.Partial well-ordering of sets of vectors. Mathematika, 1 (1954), 89–95.CrossRefGoogle Scholar
(7)Tarkowski, S.On the comparability of dendrites. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 39–41.Google Scholar