Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-01T04:50:55.142Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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