Kruskal's 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 within ATR₀ (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the 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

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 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. If , are rooted trees with vertices labeled in , we say that is inf-embeddable in and write if there is an injective 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 two distinct immediate successors of , then the path from to in contains .

Kruskal's tree theorem then states:

If is well-quasi-ordered, then the set of rooted trees with labels in is well-quasi-ordered under the inf-embeddable 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 , the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. ) such that:

* The tree at position in the sequence has no more than vertices, for all .
* 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 (where the argument specifies the number of ''labels''; see below) is larger than $\mathrm^(7)$.

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 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 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 by Π-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-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 ''T''₁,...,''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''".Smith 1985, p. 120 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 ''T''₁,...,''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 ''n''^''n''(5)(5). A lower bound for ''n''(4), and hence an ''extremely'' weak lower bound for ''TREE''(3), is ''A''^{''A''(187196)}(1), where ''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'') = 2''n'', ''A''(''k''+1, 1) = ''A''(''k'', 1), ''A''(''k''+1, ''n''+1) = ''A''(''k'', ''A''(''k''+1, ''n'')). Graham's number, for example, 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 $f_$ in the fast-growing hierarchy. ''A''^{''A''(187196)}(1) is approximately $g_$, 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_2''i'' is a subsequence of any later block x_''j'',...,x_2''j''. $n(1) = 3, n(2) = 11, \,\textrm\, n(3) > 2 \uparrow^ 158386$.

References

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{{Order theory

Mathematical logic
Order theory
Theorems in discrete mathematics
Trees (graph theory)
Wellfoundedness