# Kruskal's tree theorem

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.

## Statement

We give the version proved by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.

Given a tree $T$ with a root, and given vertices $v$ , $w$ , call $w$ a successor of $v$ if the unique path from the root to $w$ contains $v$ , and call $w$ an immediate successor of $v$ if additionally the path from to $v$ to $w$ contains no other vertex.

Take $X$ to be a partially ordered set. If $T_{1},T_{2}$ are rooted trees with vertices labeled in $X$ , we say that $T_{1}$ is inf-embeddable in $T_{2}$ and write $T_{1}\leq T_{2}$ if there is a map $F$ from the vertices of $T_{1}$ to the vertices of $T_{2}$ such that

• For all vertices $v$ of $T_{1}$ , the label of $v$ precedes the label of $F(v)$ ,
• If $w$ is any successor of $v$ in $T_{1}$ , then $F(w)$ is a successor of $F(v)$ , and
• If $w_{1},w_{2}$ are any two distinct immediate successors of $v$ , then the path from $F(w_{1})$ to $F(w_{2})$ in $T_{2}$ contains $F(v)$ .

Kruskal's tree theorem then states:

If $X$ is well-quasi-ordered, then the set of rooted trees with labels in $X$ is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence $T_{1},T_{2},\dots$ of rooted trees labeled in $X$ , there is some $i so that $T_{i}\leq T_{j}$ .)

## Weak tree function

Define tree(n), the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. $X=\{1\}$ ) such that:

• The tree at position k in the sequence has no more than k + n vertices, for all k.
• No tree is homeomorphically embeddable into any tree following it in the sequence.

It is known that tree(1) = 2, tree(2) = 5, and tree(3) > 844424930131960, but TREE(3) (see below) is larger than treetreetreetreetree8(7)(7)(7)(7)(7).

## Friedman's work

For a countable label set $X$ , 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 $X$ has order one), Friedman found that the result was unprovable in ATR0, thus giving the first example of a predicative result with a provably impredicative proof. This case of the theorem is still provable in Π1
1
-CA0, but by adding a "gap condition"  to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. Much later, the Robertson-Seymour theorem would give another theorem unprovable inside Π1
1
-CA0.

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).

### TREE(3)

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.

All the statements P(n) are true as a consequence of Kruskal's theorem and Kőnig's lemma. 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". Moreover the length of 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. 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. In fact, it is much larger than nn(5)(5). A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is AA(187196)(1), where 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 $f_{\theta (\Omega ^{\omega }\omega )}$ in the fast-growing hierarchy.