Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences. Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. Our advance here stems from a careful analysis of the Halpern-Laüchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken's tree theorem that permits us to gauge its effectivity in turn.
The principal outcome of this is a comprehensive classification of the computable content of Milliken's tree theorem. We apply our analysis also to several well-known applications of Milliken's tree theorem, namely Devlin's theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey's theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken's tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. We identify again the familiar dichotomy between coding the halting problem or not based on the size of instance, but this is more subtle here owing to the more complicated underlying structures, particularly in the case of Devlin's theorem. We also establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker's notion of big Ramsey structure.
我们研究了以下陈述的可计算性理论复杂度和证明理论强度：（1）“如果X是良序，那么φ_X也是良序”，和（2）“如果X是良序，那么φ（alpha，X）也是良序"，其中alpha是固定的可计算序数，φ是两个位置的Veblen函数。对于前一个命题，我们证明了图灵跳跃的Ω迭代在证明中是必要的，并且该命题等价于RCA_0上的ACA_0^+。为了证明后一个陈述，我们需要使用图灵跳跃的omega^alpha迭代，并且我们证明了该陈述等价于Pi^0_{omega^alpha}-CA_0。我们的证明是纯粹的计算理论。我们还对Friedman的一个结果给出了新的证明：“如果X是良序，那么phi（X，0）也是良序”等价于ATR_0 over RCA_0.
Bounded Ramsey's theorem for triples in computability theory
CNRS, Institut de Mathématiques de Jussieu–Paris Rive Gauche, Université Paris Cité（法国国家科学研究中心，儒歇-巴黎左岸数学研究所，巴黎西岱大学）；School of Mathematics, University of Leeds（利兹大学数学学院）
Computable Isomorphisms for Certain Classes of Infinite Graphs
我们研究了（2，1）：1结构，它由一个可数集$A$和一个函数$f: A \to A$组成，使得对于$A$中的每个元素$x$，$f$将$A$中的一个元素或两个元素映射到$x$ .这些结构扩展了Cenzer、Harizanov和Remmel研究的注入结构、2：1结构和（2，0）：1结构的概念，所有这些结构都可以被认为是无限有向图。我们研究了（2，1）：1结构的各种可计算性理论性质，最值得注意的是可计算范畴性。我们说一个结构$\mathcal{A}$是可计算范畴的，如果在$\mathcal{A}$的任何两个可计算副本之间存在可计算同构。给出了（2，1）：1结构可计算范畴的一个充分条件，并给出了具有不同可计算性理论性质的（2，1）：1结构的一些例子.
