Kruskal's tree theorem

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In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of Arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the notoriously fast-growing TREE function.

In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function.


We give the version proved by Nash-Williams; Kruskal's formulation is somewhat stronger. Given a tree with a root, and given vertices , , call a successor of if the unique path from the root to contains , and call an immediate successor of if additionally the path from to contains no other vertex. Take to be a partially ordered set. Taking to be rooted trees with vertices labeled in , we say that is inf-embeddable in and write if there is a map from the vertices of to the vertices of such that

  • For all vertices of , the label of precedes the label of ,
  • If is any successor of in , then is a successor of , and
  • If are any pair of distinct immediate successors of , then the path from to in contains .

Then result states that, if is well-quasi-ordered, it follows that the set of rooted trees with labels in is well-quasi-ordered under the order defined above. That is to say, given any infinite sequence of rooted trees labeled in , there is some so that .

Friedman's work[edit]

For a countable label set , Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980's, an early success of the then-nascent field of reverse mathematics; in the case where the trees above are taken to be unlabeled (that is, in the case where has order one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable in Π1
-CA0, but by adding a "gap condition" [3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson-Seymour theorem would give another theorem unprovable inside Π1

Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).


Suppose that P(n) is the statement

There is some m such that if T1,...,Tm is a finite sequence of unlabeled rooted trees where Tk has n+k vertices, then Ti ≤ Tj for some i < j.

This statement is a simple application of Kruskal's theorem, for each n, Peano arithmetic can prove that P(n) is true, but Peano arithmetic cannot prove the statement "P(n) is true for all n".[6] Moreover the shortest proof of P(n) in Peano arithmetic grows phenomenally fast as a function of n; far faster than any primitive recursive function or the Ackermann function for example. The least m for which P(n) holds similarly grows extremely quickly with n.

By incorporating labels, Friedman defined a far-faster growing function,[7] for a positive integer n, take TREE(n)[*] to be the largest m so that we have the following:

There is a sequence T1,...,Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that Ti  ≤  Tj does not hold for any i < j  ≤ m.

The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other "large" combinatorial constants, such as Friedman's n(4),[*] are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of As is A(187196),[8] and A() is a version of Ackermann's function: A(x) = 2 [x+1] x in hyperoperation. Graham's number, for example, is approximately A64(4) which is much smaller than the lower bound AA(187196)(1). It can be shown that the growth-rate of the function TREE is at least in the fast-growing hierarchy.

See also[edit]


^ * Friedman originally denoted this function by TR[n].
^ * n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[9] n(1) = 3, n(2) = 11 and n(3) > 2 [7199] 158386.


  1. ^ Simpson 1985, Theorem 1.8
  2. ^ Friedman 2002, p. 60
  3. ^ Simpson 1985, Definition 4.1
  4. ^ Simpson 1985, Theorem 5.14
  5. ^ Marcone 2001, p. 8–9
  6. ^ Smith 1985, p. 120
  7. ^ Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017. 
  8. ^ Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017. 
  9. ^ Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.