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.
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 .
Weak tree function
Define tree(n), the weak tree function, as the length of the longest sequence of 1-labelled trees such that:
- Every tree at position k (for all k) has no more than k + n vertices.
- 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) > 262140, but TREE(3) (see below) is larger than treetreetreetreetree8(7)(7)(7)(7)(7).
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, 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
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". 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.
- 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 that nn(5)(5). 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 A's is A(187196), 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.
- ^ * 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. n(1) = 3, n(2) = 11 and n(3) > 2  158386.
- Simpson 1985, Theorem 1.8
- Friedman 2002, p. 60
- Simpson 1985, Definition 4.1
- Simpson 1985, Theorem 5.14
- Marcone 2001, p. 8–9
- Smith 1985, p. 120
- Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
- Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
- 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.
- 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 (04): 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., 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., Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 119–136