Infinite Set Of Natural Numbers Research Articles

Thin set theorems and cone avoidance

The thin set theorem R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty ,\ell } asserts the existence, for every k k -coloring of the subsets of natural numbers of size n n , of an infinite set of natural numbers, all of whose subsets of size n n use at most ℓ \ell colors. Whenever ℓ = 1 \ell = 1 , the statement corresponds to Ramsey’s theorem. From a computational viewpoint, the thin set theorem admits a threshold phenomenon, in that whenever the number of colors ℓ \ell is sufficiently large with respect to the size n n of the tuples, the thin set theorem admits strong cone avoidance. Let d 0 , d 1 , … d_0, d_1, \dots be the sequence of Catalan numbers. For n ≥ 1 n \geq 1 , R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } admits strong cone avoidance if and only if ℓ ≥ d n \ell \geq d_n and cone avoidance if and only if ℓ ≥ d n − 1 \ell \geq d_{n-1} . We say that a set A A is R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable if there is an instance of R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } such that every solution computes A A . The R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable sets are precisely the hyperarithmetic sets if and only if ℓ > 2 n − 1 \ell > 2^{n-1} , the arithmetic sets if and only if 2 n − 1 ≤ ℓ > d n 2^{n-1} \leq \ell > d_n , and the computable sets if and only if d n ≤ ℓ d_n \leq \ell .

On additive representation functions

Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal to $x$. In this paper, we prove that, if $r(A, n)\not= 1$ for all sufficiently large integers $n$, then $|A(x)|> \frac 12 (\log x/\log\log x)^2$ for all sufficiently large $x$.

A Minimal Pair of Turing Degrees

Let $P(A)$ be the following property, where $A$ is any infinite set of natural numbers: \begin{displaymath} (\forall X)[X\subseteq A\wedge |A-X|=\infty\Rightarrow A\not\le_m X]. \end{displaymath} Let $({\bf R}, \le)$ be the partial ordering of all the r.e. Turing degrees. We propose the study of the order theoretic properties of the substructure $({\bf S}_{m},\le_{{\bf S}_m})$, where ${\bf S}_{m}=_{\rm dfn}\{{\bf a}\in {\bf R}$: ${\bf a}$ contains an infinite set $A$ such that P(A) is true$\}$, and $\le_{{\bf S}_m}$ is the restriction of $\le$ to ${\bf S}_m$. In this paper we start by studying the existence of minimal pairs in ${\bf S}_{m}$.

Homogeneous sets from several ultrafilters

We consider Ramsey-style partition theorems in which homogeneity is asserted not for subsets of a single infinite homogeneous set but for subsets whose elements are chosen, in a specified pattern, from several sets in prescribed ultrafilters. We completely characterize the sequences of ultrafilters satisfying such partition theorems. (Non-isomorphic selective ultrafilters always work, but, depending on the specified pattern, weaker hypotheses on the ultrafilters may suffice.) We also obtain similar results for analytic partitions of the infinite sets of natural numbers. Finally, we show that the two P-points obtained by applying the maximum and minimum functions to a union ultrafilter are never nearly coherent.

$${\Pi^1_2}$$ -comprehension and the property of Ramsey

We show that a theory of autonomous iterated Ramseyness based on second order arithmetic (SOA) is proof-theoretically equivalent to $${\Pi^1_2}$$ -comprehension. The property of Ramsey is defined as follows. Let X be a set of real numbers, i.e. a set of infinite sets of natural numbers. We call a set H of natural numbers homogeneous for X if either all infinite subsets of H are in X or all infinite subsets of H are not in X. X has the property of Ramsey if there exists a set which is homogeneous for X. The property of Ramsey is considered in reverse mathematics to compare the strength of subsystems of SOA. To characterize the system of $${\Pi^1_2}$$ -comprehension in terms of Ramseyness we introduce a system of autonomous iterated Ramseyness, called R-calculus. We augment the language of SOA with additional set terms (called R-terms) $${R\vec{x}X\phi(\vec{x},X)}$$ for each first order formula $${\phi(\vec{x},X)}$$ (where φ may contain further R-terms). The R-calculus is a system which comprises comprehension for all first order formulas (which may contain R-terms or other set parameters) and defining axioms for the R-terms which claim that for each $${\vec{x}}$$ , we can remove finitely many elements from the set $${R\vec{x}X\phi(\vec{x},X)}$$ such that the remaining set is homogeneous for $${\{{X}{\phi(\vec{x},X)\}}}$$ . We show that the R-calculus proves the same $${\Pi^1_1}$$ -sentences as the system of $${\Pi^1_2}$$ -comprehension.

Parameterized partition relations on the real numbers

We consider several kinds of partition relations on the set \({\mathbb{R}}\) of real numbers and its powers, as well as their parameterizations with the set \({[\mathbb{N}]^{\mathbb{N}}}\) of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of \({\mathbb{R}^\mathbb{N}}\) there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of \({[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}\) there is \({X \in [\mathbb{N}]^{\mathbb{N}}}\) and a sequence \({\{P_{k} : k \in \mathbb{N}\}}\) of perfect sets such that the product \({[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}\) lies in one piece of the partition, where \({[X]^{\mathbb{N}}}\) is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness.

A VARIANT ON THE NOTION OF A DIOPHANTINE s-TUPLE

AbstractWe show that there is an infinite set of natural numbers with the property that $1+\prod_{n\in\cR}n$ is square-free for every finite subset ⊆ .

Doughnuts, floating ordinals, square brackets, and ultraflitters

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

On the Distribution ofB3-Sequences

An infinite set of natural numbers is called aB3-sequence if all sumsa1+a2+a3withaj∈Aanda1⩽a2⩽a3are distinct. LetA(n) be the number of positive elements ⩽ninA. P. Erdös conjectures that everyB3-sequenceAsatisfies liminfn→∞A(n) n−1/3=0. In this paper we prove that no sequence satisfyingA(n)∼αn1/3can be aB3-sequence. We also give other necessary conditions for aB3-sequence.

Effective versions of Ramsey's Theorem: Avoiding the cone above 0′

AbstractRamsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a set A. Applications to reverse mathematics and introreducible sets are discussed.

Congruent subsets of infinite sets of natural numbers.

This improves an earlier result of the first author for k = n = 2, namely Theorem II below, which dealt with B2 -sequences. This notion arose from a paper of Sidon [6]. A strictly increasing sequence A of natural numbers a,, a2 . . . . is called a B2-sequence, if for any two pairs (i, j) + (k, m) of natural numbers i <j, k < m there holds aj a i + ak a„„ in short, if A has no double-differences . (Of course it would be the same condition to presuppose that A has no double-sums : For any two pairs (i, j) + (k, m) with i < j, k < m there holds ai + a, + a k + a,,, .)

Determinateness and partitions

It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that ω → ( ω ) 2 ω \omega \to (\omega )_2^\omega (i.e. for every partition of infinite sets of natural numbers into two classes there is an infinite set such that all its infinite subsets belong to the same class).

Determinateness and Partitions

It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that $\omega \to (\omega )_2^\omega$ (i.e. for every partition of infinite sets of natural numbers into two classes there is an infinite set such that all its infinite subsets belong to the same class).

On the difficulty of computations

Two practical considerations concerning the use of computing machines are the amount of information that must be given to the machine for it to perform a given task and the time it takes the machine to perform it. The size of programs and their running time are studied for mathematical models of computing machines. The study of the amount of information (i.e., number of bits) in a computer program needed for it to put out a given finite binary sequence leads to a definition of a random sequence; the random sequences of a given length are those that require the longest programs. The study of the running time of programs for computing infinite sets of natural numbers leads to an arithmetic of computers, which is a distributive lattice.

A note on degrees of subsets1

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers

It is suggested that there are infinite computable sets of natural numbers with the property that no infinite subset can be computed more simply or more quickly than the whole set. Attempts to establish this without restricting in any way the computer involved in the calculations are not

Sets with no subset of higher degree

The problem of finding an infinite set of natural numbers which contains no subsets of higher (Turing) degree was first posed by W. Miller [3] and was brought to our attention by C. G. Jockusch, Jr., who proved that such a set, if it existed, could not be hyperarithmetic.2 In this paper we construct an infinite set which is not recursive in any of its coinfinite subsets, and thus contains no subset of higher degree. Our original proof made use of the result (attributed to Ehrenfeucht) that every subset of 2ω which is open (in the standard topology) is “Ramsey”.