We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2,2) nor OVW(2,2) is provable in WKL 0 . These results give partial answers (...) to questions posed by Friedman and Simpson [3]. (shrink)

We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for (...) barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

Solovay has shown that if F: [ω] ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0 α , where α is a recursive ordinal, there is a clopen partition F: [ω] ω → 2 such that every infinite homogeneous set is Turing above 0 α (an anti-basis result). Here we refine these results, by associating the "order (...) type" of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems (...) for equivalence relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are nonnull in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for nonnull many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree (...) below $\emptyset'$ but not in general. We also answer the analogs of two well-known questions about the stable Ramsey's theorem by showing that our weaker principle does not imply COH or WKL 0 in the context of reverse mathematics. (shrink)

Ramsey’s theorem asserts that every k-coloring of $[\omega ]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable k-coloring of $[\omega ]^n$ whose solutions compute the halting set. On the other hand, for every computable k-coloring of $[\omega ]^2$ and every noncomputable set C, there is an infinite monochromatic set H such that $C \not \leq _T H$. The latter property is known as cone avoidance.In this article, we design a natural class (...) of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every k-coloring of $[\omega ]^n$, of an infinite subdomain $H \subseteq \omega $ over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) (...) . Let IΣ n and BΣ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that (...) every computable 2-coloring of pairs admits a pair of low₂ infinite homogeneous sets whose degrees form a minimal pair. (shrink)

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and (...) some branch with infinitely many children of indexi. (shrink)

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey's Theorem for Pairs is not strong enough to prove , the comprehension scheme for arithmetical formulas, within the base theory , the comprehension scheme (...) for recursive formulas. We also show that Ramsey's Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within . In particular, Ramsey's Theorem for Pairs is not conservative over for -sentences. (shrink)

Let X be an infinite internal set in an ω1-saturated nonstandard universe. Then for any coloring of [X] k , such that the equivalence E of having the same color is countably determined and there is no infinite internal subset of [X] k with all its elements of different colors (i.e., E is condensating on X), there exists an infinite internal set Z⊆X such that all the sets in [Z] k have the same color. This Ramsey-type (...) result is obtained as a consequence of a more general one, asserting the existence of infinite internal Q-homogeneous sets for certain Q ⊆ [[X] k ] m , with arbitrary standard k≥ 1, m≥ 2. In the course of the proof certain minimal condensating countably determined sets will be described. (shrink)

In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A all of whose size n subsets are contained in only one of X or Y.

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is (...) well-known that Ramsey's Theorem for pairs (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)

Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we (...) prove that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. (shrink)

It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible (...) in various senses. (shrink)

It is known that for two given countable sets of unary relations A and B on ω there exists an infinite set H ⫅ ω on which A and B are the same. This result can be used to generate counterexamples in expressibility theory. We examine the sharpness of this result.

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of (...) large-small filters requires a nonconstructive axiom like AC. (shrink)

Let G be a group, and let X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of P in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X. We describe all G-spaces X (...) such that each free ultrafilter on X is G-selective, and we prove that a free ultrafilter U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω. A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ:[X]2→{0,1}, there is U∈U such that [U]2 is χ-monochromatic. We show that each G-Ramsey ultrafilter on X is G-selective. Additional theorems give a lot of examples of ultrafilters on Z that are Z-selective but not Z-Ramsey. (shrink)

For a function $f$ with domain $[X]^{n}$, where $X\subseteq\mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon\subseteq n$ such that for any $x_{0},\ldots,x_{n-1}$ and $y_{0},\ldots,y_{n-1}$ in $H$, $f=f$ iff $x_{i}=y_{i}$ for all $i\in\upsilon$. The canonical Ramsey theorem is the statement that for any $n\in\mathbb{N}$, if $f:[\mathbb{N}]^{n}\rightarrow\mathbb{N}$, then there is an infinite $H\subseteq\mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey (...) theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $\operatorname{PA}$. (shrink)

Ramsey type theorems are theorems of the form: if certain sets are partitioned at least one of the parts has some particular property. In its finite form, Ramsey's theory will ask how big the partitioned set should be to assure this fact. Proofs of such theorems usually require a process of multiple choice, so that this apparently pure combinatoric field is rich in proofs that use ideal guides in making the choices. Typically they may be ultrafilters or points (...) in the compactification of the given set. It is, therefore, not surprising that nonstandard elements are much more natural guides in some of the proofs and in the general abstract treatment.In Section 1 we start off with some very natural examples of Ramsey type exercises that illustrate our idea. In Section 2 we give a nonstandard proof of the infinite Ramsey theorem. Section 3 tries to do the same for Hindman's theorem, and points out, where nonstandard analysis must use some hard standard facts to make the proof go through. (shrink)

No chance of seeing her for another fortnight and it is 11 days since I saw her. Went solitary walk felt miserable but to some extent staved it off by reflecting on |$\langle$|Continuum Problem|$\rangle$|1The occasion for this review article on the life and accomplishments of Frank Ramsey is the publication in the last eight years of three important books: a biography of Frank Ramsey by his sister, Margaret Paul, a book by Steven Methven on aspects of Ramsey’s (...) philosophy, and the recent biography of Ramsey written by Cheryl Misak. In a companion piece [Mancosu, 2020] I discuss Ramsey’s contacts with logicians in continental Europe and publish previously unpublished letters from Ramsey to Abraham Fraenkel and to Heinrich Behmann.1. INTRODUCTIONFrank Plumpton Ramsey (1903–1930) will be well known to the readers of this journal. His contributions to philosophy, mathematics, and economics have left a lasting legacy. In mathematical logic and philosophy of mathematics, Ramsey’s solution to the paradoxes by means of a simple theory of types [1926a] and Ramsey’s theorem in combinatorics (both finite and infinite [1930]) are among Ramsey’s most celebrated contributions. Ramsey is a key presence in accounts of Wittgenstein’s, Russell’s, and Moore’s lives and work. It is well known that it was Ramsey who translated the Tractatus into English and wrote what is still considered a very insightful review [1923] of the Tractatus. Wittgenstein mentioned the importance of his conversations with Ramsey in 1929 in Cambridge as having influenced his transition to the views presented in the Philosophical Investigations. Ramsey also contributed, in 1924, to the second edition of Principia Mathematica (see [Linsky, 2011]). While Ramsey’s friendship with Moore does not have an analogous concrete precipitate as the work on the Tractatus or Principia Mathematica, it is evident from the correspondence and other documents that Ramsey’s work in philosophy (truth, propositions, universals, etc.) influenced Moore. Of course, the direction of influence goes also the other way around. Wittgenstein, Russell, and Moore, among others, were essential in shaping Ramsey’s approach to philosophy. (shrink)

We analyze three applications of Ramsey’s Theorem for 4-tuples to infinite traceable graphs and finitely generated infinite lattices using the tools of reverse mathematics. The applications in graph theory are shown to be equivalent to Ramsey’s Theorem while the application in lattice theory is shown to be provable in the weaker system RCA0.

We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L ∞,ω to L κ +, ω . The simplest instance is: Theorem 1. Let $\lambda > \kappa$ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every $X \subseteq M$ there exists a model $N \prec M$ containing the set X of power |X| · κ such that for every pair of (...) finite sequences a, b ∈ N $\langle N, \mathbf{a}\rangle \equiv_{\| N \|^+,\omega} \langle N, \mathbf{b}\rangle \Leftrightarrow \langle M, \mathbf{a}\rangle \equiv_{\infty,\omega} \langle M, \mathbf{b}\rangle.$ The following theorem is an application: Theorem 2. Let $\lambda , and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, L κ +, ω -unsuperstability property, then T has the (χ, L λ +, ω )-unsuperstability property. (shrink)

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K>1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K>1, K-bi-Lipschitz embed into the real line.

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.

Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.

Let AC and AI denote the collections of graphs with vertex set ω and which have, respectively, no infinite independent subgraph, and no infinite complete subgraph. Both AC and AI are coanalytic sets of reals that are not analytic, and they are disjoint by Ramsey’s theorem. We prove that there exists a Borel set separating AC and AI, and we discuss the sense in which this is an infinite analogue of a weak version of.

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and (...) strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements. (shrink)

The thin set theorem for n-tuples and k colors states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$, nor the free set theorem imply the Erdős–Moser theorem whenever k is sufficiently large. Given a problem $\mathsf {P}$, a computable instance (...) of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance. (shrink)

We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically saturated models of Peano (...) Arithmetic such that Aut(M1)≅ Aut(M2).ThenSSy(M1) = SSy(M2).We show that ifMis a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1,M2be countable arithmetically saturated models of Peano Arithmetic such thatAut(M1) ≅ Aut(M2).Then for every n<ωHere RT2nis Infinite Ramsey's Theorem stating that every 2-coloring of [ω]nhas an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic. (shrink)

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be (...) strictly stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$. (shrink)

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow (...) class='Hi'>Ramsey Theorem is not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

The Rainbow Ramsey Theorem is essentially an "anti-Ramsey" theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA₀ of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey's (...) Theorem for pairs over RCA₀. The separation involves techniques from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin's theorem that the hyperimmune degrees have measure one. (shrink)

A set of integers A is computably encodable if every infinite set of integers has an infinite subset computing A. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this article, we extend this notion of computable encodability to subsets of the Baire space, and we characterize the Π10-encodable compact sets as those which admit a nonempty Σ11-subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly computably (...) reducible to Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch. (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only (...) use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature (...) are indeed [Formula: see text]. (shrink)

Frank Ramsey's ‘Truth and Probability’ sketches a proposal for the empirical measurement of credences, along with a corresponding set of axioms for a representation theorem intended to characterize the preference conditions under which this measurement process is applicable. There are several features of Ramsey's formal system which make it attractive and worth developing. However, in specifying his measurement process and his axioms, Ramsey introduces the notion of an ethically neutral proposition, the assumed existence of which plays (...) a key role throughout Ramsey's system. A number of later representation theorems have also appealed to ethically neutral propositions. The notion of ethical neutrality has often been called into question — in fact, there seem to be good reasons to suppose that no ethically neutral propositions exist. In this paper, I present several new, Ramsey-inspired representation theorems that avoid any appeal to ethical neutrality. These theorems preserve the benefits of Ramsey's system, without paying the cost of ethical neutrality. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.3 (2001) 483-503 [Access article in PDF] China and the Ideal of Order in John Webb's An Historical Essay.... Rachel Ramsey Scholars of seventeenth-century intellectual history have generally relegated John Webb to the footnotes of their work on universal language schemes, architectural history, and Sino-European relations. 1 In this essay I suggest that Webb's An Historical Essay Endeavoring a Probability that the (...) Language of the Empire of China is the Primitive Language (1669), which argues that for 5000 years China preserved the language spoken by Adam and Eve in the Garden of Eden, makes significant contributions to seventeenth-century intellectual history and to our understanding of early modern European perceptions of China. When Webb's Essay is placed within the context of the seventeenth-century debates about the "primitive" language of Eden, China's ancient history, and the idealization of that empire's harmonious and prosperous culture, its deeply political nature becomes apparent.The Jesuit accounts and those of other travelers to China were especially appealing to English readers living in the aftermath of the civil wars and the turmoil of the early years of the Restoration because they offered glimpses of a seemingly ideal state ruled by a stable monarchy and blessed with seemingly infinite resources and unimaginable wealth; however, these accounts included evidence that China had maintained an unbroken historical record that antedated the Flood, thus implicitly challenging the veracity of the Old Testament account of Noah and the Universal Deluge. Webb's Essay solves the problem posed by the Jesuit accounts of China's ancient history by placing them within a revised biblical narrative; this solution makes it possible for him to claim that China's [End Page 483] exemplary status is a result of its possession of the primitive language and its socio-political virtues are the consequence of its Noachian origins. The revised biblical narrative not only made Webb's encomium to that country more acceptable to a bible-reading public and protestant king but offered a politically-safe way for him to criticize the restored English monarchy by contrasting it to an idealized account of China's social and political state. More specifically, the circumstances of Webb's years of service in the Office of the King's Works help us further understand what attracted a seventeenth-century architect to the Jesuit accounts of China and what motivated him to write the first English treatise on the Chinese language. Webb's political career allows us to read AnHistoricalEssay as a politico-theological justification for a reasoned critique of the patronage system which Webb held accountable for his thwarted career ambitions. Webb's Career and the Politics of the Restoration John Webb lived through some of England's most turbulent history and witnessed firsthand the effects of monarchical instability and regicide, the uncertainties of the Commonwealth, and the shortcomings of the Restoration. Born to a Somerset family in 1611 and educated at the Merchant Taylors' school, Webb became Inigo Jones's pupil in 1628 and served as his "Clerk Engrosser" from 1633 to 1641 in the Office of the King's Works, where Jones was Surveyor. During the early years of the English revolution Webb served as his mentor's deputy in London after Jones fled north with Charles I, but he was dismissed from his post in 1643 following an "accusation from one Mr. Carter to the Committee of the Revenue, that the said Mr. Jones was at Oxford." 2 During his absence from the Office of the Works, Webb supposedly sent detailed plans of London's fortifications and smuggled jewelry to the king, for which he was briefly imprisoned. 3 After Charles I's execution in 1649, Webb worked on several country houses, including those of the Earl of Rutland at Belvoir, the Earl of Peterborough at Drayton, and Sir Justinian Isham at Lamport but he was firmly excluded from holding any political post as a former consort of the martyred King. 4 He held an appointment briefly... (shrink)

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive (...) recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and (...) RT[Formula: see text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman. (shrink)

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q = {Qi: iε I} is a family (...) of generalized quantifiers on finite structures. Using the logics L∞ωω, we can express polynomial-time properties that are not definable in L∞ωω, such as “there is an even number of x” and “there exists at least n/2 x” , without going to second-order logic. We show that equivalence of finite structures relative to L∞ωω can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L∞ωω and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L∞ωω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L∞ωω. We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L∞ωω of L∞ωω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers. (shrink)

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.