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 Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). 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 T₁, T₂ are rooted trees with vertices labeled in X, we say that T₁ is inf-embeddable in T₂ and write $T_{1}\leq T_{2}$ if there is an injective map F from the vertices of T₁ to the vertices of T₂ such that:

• For all vertices v of T₁, the label of v precedes the label of $F(v)$;
• If w is any successor of v in T₁, then $F(w)$ is a successor of $F(v)$; and
• If w₁, w₂ are any two distinct immediate successors of v, then the path from $F(w_{1})$ to $F(w_{2})$ in T₂ 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 T₁, T₂, … 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 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 by 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).

Weak tree function

Suppose that $P(n)$ is the statement:

There is some m such that if T₁, ..., 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 T₁, ..., 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.

TREE function

A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.

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

There is a sequence T₁, ..., T_m of rooted trees labelled from a set of n labels, where each T_i has at most i 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,^[b] are extremely small by comparison. A lower bound for $n(4)$, and, hence, an extremely weak lower bound for ${\text{TREE}}(3)$, is $A^{A(187196)}(1)$.^[c] 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.

See also

• Paris–Harrington theorem
• Kanamori–McAloon theorem
• Robertson–Seymour theorem

Notes

^{^ a} Friedman originally denoted this function by TR[n].

^{^ b} 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 x_i,...,x_2i is a subsequence of any later block x_j,...,x_2j. $n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386$.

^{^ c} A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).

References

Citations

Bibliography

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