克鲁斯卡尔树定理

TREE函数、Kruskal树定理（Kruskal's tree theorem）是逆数学的突出示例。由安德鲁·瓦兹尼推测并由约瑟夫·克鲁斯卡尔证明。
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.
在数学中，克鲁斯卡树定理指出，在准有序（英文：well-quasi-ordering）标签集上的有限树集合本身在同胚下也是良准有序的（英文：同胚（图论））嵌入。
历史
该定理由安德鲁·瓦兹索尼（Andrew Vázsonyi）猜想并由约瑟夫·克鲁斯卡尔（Joseph Kruskal，1960）证明； Crispin Nash-Williams（1963）给了一个简短的证明。它从此成为逆向数学（英语：reverse math）中的一个突出例子，作为在 ATR0 内无法证明的陈述（算术超限递归（英语：arithmetic transfinite recursion）的一种形式）和有限（英语：finitary）应用该定理给出了快速增长的TREE 函数的存在性。
2004年，这个结果被从树推广到图，成为罗伯逊-西摩定理（英语：Robertson-Seymour theorem），这一结果在逆向数学中也被证明很重要，并导致了增长更快的SSCG 函数（英语： Friedman 的 SSCG 函数）。
陈述
这里给出的版本是由 Nash-Williams 证明的；克鲁斯卡尔的表述要强一些。我们认为的所有树木都是有限的。
给定一棵有根的树T，并给定顶点v、w，如果从根到w 的唯一路径包含v，则称w 为v 的后继；如果从v 到w 的路径另外不包含v，则称w 为v 的直接后继其他顶点。
取X为偏序集（中文：partiallyorderedset）。如果 T1、T2 是顶点标记为 X 的有根树，我们说 T1 可 inf-embeddable 在 T2 中，并且如果存在从 T1 的顶点到 T2 的顶点的单射映射 F，则写 T1 ≤ T2，使得
弗里德曼的工作
对于可数标签集
X
X.克鲁斯卡树定理可以用二阶算术来表达和证明（中文：二阶算术）。然而，就像古德斯坦定理或巴黎-哈灵顿定理（英语：Paris-Harrington theorem）一样，该定理的一些特殊情况和变体可以用比可以证明它们的子系统弱得多的二阶算术子系统来表达。这是哈维·弗里德曼（Harvey Friedman）在 20 世纪 80 年代初首次观察到的，这是当时新兴的逆向数学领域的早期成功。在上面的树被认为是未标记的情况下（即，在
X
X 有顺序一），Friedman 发现结果在 ATR0（英语：算术超限递归）,2 从而给出了第一个谓词（英语：impredicativity）结果的例子，并带有可证明的谓词证明.3此定理的这种情况仍可由 Π1 证明
1-CA0，但透过在上述树的顺序定义中添加「间隙条件」4，他发现该定理的自然变体在该系统中无法证明。56很久以后，罗伯森-西摩定理给了另一个无法由 Π1 证明的定理
1-CA0。
序数分析（中文：序数分析）证实了克鲁斯卡尔定理的强度，该定理的证明论序数等于小维勃伦序数（英文：小维勃伦序数）（有时与较小的阿克曼序数（英文：阿克曼序数）相混淆） ).来源请求
树(3)
假设 P(n) 是这样的语句：
存在一些 m，如果 T1,...,Tm 是未标记有根树的有限序列，其中 Tk 有 n+k 个顶点，则对于某些 i 0 (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 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 T1 ≤ T2 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(w1) to F(w2) 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 i ≤ Tj.)

**Weak tree function**
Define tree(n), the weak tree function, as the length of the longest sequence of 1-labelled trees (i.e. ) 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) = 5, and tree(3) ≥ 844424930131960,reason=WP:RSPSOURCES; Fandom page lists YT video as reference but TREE(3) (where the argument specifies the number of labels; see below) is larger than 

**Friedman's work**
For a countable label set , Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like 古德斯坦定理 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  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 Πb=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 Πb=1-CA0.
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).date=September 2022

**TREE(3)**
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 1,...,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 n(5)(5). A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is AA(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) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)). 葛立恒数, for example, 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. AA(187196)(1) is approximately , where gx is Graham's function.

**参见**
Paris–Harrington theorem
*Kanamori–McAloon theorem
*Robertson–Seymour theorem

**注释**
^ * Friedman originally denoted this function by TRn.
^ ** 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. .

**参考**
Citations
Bibliography
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