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

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

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

Kruskal's tree theorem then states:

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

Weak tree function

Define ${\displaystyle \mathrm {tree} (n)}$, the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. ${\displaystyle X=\{1\}}$) 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) = 3, and tree(3) > 844424930131960,[1] but ${\displaystyle \mathrm {TREE} (3)}$ (see below) is larger than ${\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7)}$.

Friedman's work

For a countable label set ${\displaystyle X}$, 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 ${\displaystyle X}$ has order one), Friedman found that the result was unprovable in ATR₀,[2] thus giving the first example of a predicative result with a provably impredicative proof.[3] This case of the theorem is still provable in Π¹
₁-CA₀, but by adding a "gap condition"[4] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[5][6] 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,...,T_m is a finite sequence of unlabeled rooted trees where T_k has n+k vertices, then T_i ≤ 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".[7] 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.[8] For a positive integer n, take TREE(n)^[*] 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  ≤  T_j 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 immensely 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 A^A(187196)(1),[9] where A() is a version of Ackermann's function: ${\displaystyle A(x)=2\uparrow ^{x-1}x}$. Graham's number, for example, is approximately A⁶⁴(4), which is much smaller than the lower bound A^A(187196)(1). It can be shown that the growth-rate of the function TREE is at least ${\displaystyle f_{\theta (\Omega ^{\omega }\omega )}}$ in the fast-growing hierarchy. A^A(187196)(1) is approximately ${\displaystyle g_{3\uparrow ^{187196}3}}$, where g_x is Graham's function.

See also

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

Notes

^{^ *} 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 x_i,...,x_2i is a subsequence of any later block x_j,...,x_2j.[10] ${\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}$.

References

Citations

1. "TREE sequence". Googology Wiki | Fandom. Retrieved 9 July 2020.
2. Simpson 1985, Theorem 1.8
3. Friedman 2002, p. 60
4. Simpson 1985, Definition 4.1
5. Simpson 1985, Theorem 5.14
6. Marcone 2001, p. 8–9
7. Smith 1985, p. 120
8. Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
9. Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
10. 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.

Bibliography

• Friedman, Harvey M. (2002), Internal finite tree embeddings. Reflections on the foundations of mathematics (Stanford, CA, 1998), Lect. Notes Log., 15, Urbana, IL: Assoc. Symbol. Logic, pp. 60–91, MR 1943303
• Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory" (PDF), Ann. Pure Appl. Logic, 53 (3): 199–260, doi:10.1016/0168-0072(91)90022-E, MR 1129778
• Kruskal, J. B. (May 1960), "Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture" (PDF), Transactions of the American Mathematical Society, American Mathematical Society, 95 (2): 210–225, doi:10.2307/1993287, JSTOR 1993287, MR 0111704
• Marcone, Alberto (2001). "Wqo and bqo theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. 21: 303–330.
• Nash-Williams, C. St.J. A. (1963), "On well-quasi-ordering finite trees", Proc. Camb. Phil. Soc., 59 (4): 833–835, doi:10.1017/S0305004100003844, MR 0153601
• Rathjen, Michael; Weiermann, Andreas (1993). "Proof-theoretic investigations on Kruskal's theorem". Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-g.
• Simpson, Stephen G. (1985), "Nonprovability of certain combinatorial properties of finite trees", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 87–117
• Smith, Rick L. (1985), "The consistency strengths of some finite forms of the Higman and Kruskal theorems", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 119–136