scholarly journals THE TOPOLOGICAL PIGEONHOLE PRINCIPLE FOR ORDINALS

2016 ◽  
Vol 81 (2) ◽  
pp. 662-686 ◽  
Author(s):  
JACOB HILTON
Keyword(s):  
Order Type ◽  
Pigeonhole Principle ◽  
Correct Order ◽  
Partition Relation ◽  
Independence Result ◽  
Homogeneous Set ◽  
Image Position

AbstractGiven a cardinal κ and a sequence ${\left( {{\alpha _i}} \right)_{i \in \kappa }}$ of ordinals, we determine the least ordinal β (when one exists) such that the topological partition relation$$\beta \to \left( {top\,{\alpha _i}} \right)_{i \in \kappa }^1$$holds, including an independence result for one class of cases. Here the prefix “top” means that the homogeneous set must be of the correct homeomorphism class rather than the correct order type. The answer is linked to the nontopological pigeonhole principle of Milner and Rado.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

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 if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is 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 that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has 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.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
Author(s):  
SAMUEL R. BUSS ◽  
LESZEK ALEKSANDER KOŁODZIEJCZYK ◽  
NEIL THAPEN
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

An ordinal partition avoiding pentagrams

2000 ◽  
Vol 65 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Jean A. Larson
Keyword(s):  
Independent Set ◽  
Order Type ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Image Position

AbstractSuppose that α = γ + δ where γ ≥ δ > 0. Then there is a graph which has no independent set of order type and has no pentagram (a pentagram is a set of five points with all pairs joined by edges). In the notation of Erdős and Rado. who generalized Ramsey's Theorem to this setting.

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

Weak partition relations and measurability

1986 ◽  
Vol 51 (1) ◽  
pp. 33-38
Author(s):  
Mitchell Spector
Keyword(s):  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinal ◽  
Partition Relation ◽  
Consistency Strength ◽  
Partition Relations ◽  
Inner Model ◽  
Increasing Function ◽  
Natural Way

The concept of "partition relation" has proven to be extremely important in the development of the theory of large cardinals. This is due in good part to the fact that the ordinal numbers which appear as parameters in partition relations provide a natural way to define a detailed hierarchy of the corresponding large cardinal axioms. In particular, the study of cardinals satisfying Ramsey-Erdös-style partition relations has yielded a great number of very interesting large cardinal axioms which lie in strength strictly between inaccessibility and measurability. It is the purpose of this paper to show that this phenomenon does not occur if we use infinite exponent partition relations; no such partition relation has consistency strength strictly between inaccessibility and measurability. We also give a complete determination of which infinite exponent partition relations hold, assuming that there is no inner model of set theory with a measurable cardinal.Our notation is standard. If F is a function and x is a set, then F″x denotes the range of F on x. If X is a set of ordinals and α is an ordinal, then [X]α is the collection of all subsets of X of order type α. We identify a member of [X]α with a strictly increasing function from α to X. If p ∈ [X]α and q ∈ [α]β, then the composition of p with q, which we denote pq, is a member of [X]β.

An extremal function for contractions of graphs

1984 ◽  
Vol 95 (2) ◽  
pp. 261-265 ◽  
Author(s):  
Andrew Thomason
Keyword(s):  
Random Graph ◽  
Extremal Function ◽  
Correct Order ◽  
Simple Argument ◽  
Image Position ◽  
Positive Integers

AbstractThe function c(p) is defined for positive integers p ≥ 4 bywhere > denotes contraction. Random graph examples showIn 1968 Mader showed that c(p) ≤ 8(p − 2) [log2 (p − 2)] and more recently Kostochka showed that p√(log p) is the correct order for c(p). We present a simple argument showing c(p) ≤ 2.68p √(log2p)(l + ο(l)).

On large cardinals and partition relations

1971 ◽  
Vol 36 (2) ◽  
pp. 305-308 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Partition Relation ◽  
Weakly Compact ◽  
Original Result ◽  
Partition Relations ◽  
Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

A recursion theoretic analysis of the clopen Ramsey theorem

1984 ◽  
Vol 49 (2) ◽  
pp. 376-400 ◽  
Author(s):  
Peter Clote
Keyword(s):  
Order Type ◽  
Theoretic Analysis ◽  
Ramsey Theorem ◽  
Homogeneous Set

AbstractSolovay 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 0α, where α is a recursive ordinal, there is a clopen partition F: [ω]ω → 2 such that every infinite homogeneous set is Turing above 0α (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.

The consistency of one fixed omega

1995 ◽  
Vol 60 (1) ◽  
pp. 172-177 ◽  
Author(s):  
J. M. Henle
Keyword(s):  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Image Position ◽  
The Way

AbstractThe paper “Partitions of Products” [DiPH] investigated the polarized partition relationThe relation is consistent relative to an inaccessible cardinal if every αi is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite.Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω.

scholarly journals LOWER BOUNDS FOR DNF-REFUTATIONS OF A RELATIVIZED WEAK PIGEONHOLE PRINCIPLE

2015 ◽  
Vol 80 (2) ◽  
pp. 450-476 ◽  
Author(s):  
ALBERT ATSERIAS ◽  
MORITZ MÜLLER ◽  
SERGI OLIVA
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Pigeonhole Principle ◽  
Low Degree ◽  
Robustness Condition ◽  
Image Position

AbstractThe relativized weak pigeonhole principle states that if at least 2n out of n2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size $2^{\left( {{\rm{log\ }}n} \right)^{3/2 - \varepsilon } } $ for every ε﹥0 and every sufficiently large n. By reducing it to the standard weak pigeonhole principle with 2n pigeons and n holes, we also show that this lower bound is essentially tight in that there exist DNF-refutations of size $2^{\left( {{\rm{log\ }}n} \right)^{O\left( 1 \right)} } $ even in R(log). For the lower bound proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.

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