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.

History
The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by. It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).

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, which dwarfs $$\text{TREE}(3)$$. A finitary application of the theorem gives the existence of the fast-growing TREE function.

Statement
The version given here is that proven 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 $v$ 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 an injective 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 $T_{1}, T_{2}, …$ of rooted trees labeled in $X$, there is some $i<j$ so that $T_i \leq T_j$.)"

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 1980s, 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 size 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 by Π$1 1$-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 by Π$1 1$-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).

Weak tree function
Suppose that $$P(n)$$ is the statement:


 * There is some $m$ such that if $T_{1}, ..., T_{m}$ is a finite sequence of unlabeled rooted trees where $T_{i}$ has $$i+n$$ vertices, then $$T_i \leq T_j$$ 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 $n$.

Define $$\text{tree}(n)$$, the weak tree function, as the largest $m$ so that we have the following:


 * There is a sequence $T_{1}, ..., T_{m}$ of unlabeled rooted trees, where each $T_{i}$ has at most $$i+n$$ vertices, such that $$T_i \leq T_j$$ does not hold for any $$i<j\leq m$$.

It is known that $$\text{tree}(1)=2$$, $$\text{tree}(2)=5$$, $$\text{tree}(3) \geq 844,424,930,131,960$$ (about 844 trillion), $$\text{tree}(4) \gg g_{64}$$ (where $$g_{64}$$ is Graham's number), and $$\text{TREE}(3)$$ (where the argument specifies the number of labels; see below) is larger than $$\mathrm{tree}^{\mathrm{tree}^{\mathrm{tree}^{\mathrm{tree}^{\mathrm{tree}^{8}(7)}(7)}(7)}(7)}(7).$$

To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.

Weak TREE function history
Believed to be started by a random forum poster online, the weak tree function is a more "restricted" form of the TREE function in which while numbers will still grow to unimagniable proportions, it will not be as big as the regular tree function.

TREE function


By incorporating labels, Friedman defined a far faster-growing function. For a positive integer $TREE(3)$, take $$\text{TREE}(n)$$ to be the largest $n$ so that we have the following:


 * There is a sequence $m$ of rooted trees labelled from a set of $n$ labels, where each $n$ has at most $n$ vertices, such that $$T_i \leq T_j$$ does not hold for any $$i<j\leq m$$.

The TREE sequence begins $$\text{TREE}(1)=1$$, $$\text{TREE}(2)=3$$, then suddenly, $$\text{TREE}(3)$$ explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's $$n(4)$$, $$n^{n(5)}(5)$$, and Graham's number, are extremely small by comparison. A lower bound for $$n(4)$$, and, hence, an extremely weak lower bound for $$\text{TREE}(3)$$, is $$A^{A(187196)}(1)$$. Graham's number, for example, is much smaller than the lower bound $$A^{A(187196)}(1)$$, which is approximately $$g_{3 \uparrow^{187196} 3}$$, where $$g_x$$ is Graham's function.