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. In this sense, Milliken's tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive.Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey's theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken's tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-La�uchli theorem, which is a key ingredient.Our advance here stems from a careful analysis of the Halpern-L�uchli 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 key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken's tree theorem. This enables us to build solutions to the full Milliken's tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken's tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove 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. In particular, we 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.

N� de r�f. du vendeur LU-9781470467319

Contacter le vendeur

Signaler cet article

D�tails bibliographiques

Titre: Milliken's Tree Theorem and Its Applications: a Computability-Theoretic Perspective
�diteur: American Mathematical Society, US
Date de parution: 2024
Langue: anglais
ISBN � 10 chiffres: 1470467313
ISBN � 13 chiffres: 9781470467319
Reliure: Paperback
�tat de l'article: New
Poids: 272�grammes

A propos de ce titre

Synopsis

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. In this sense, Milliken's tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive.

Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey's theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken's tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-La�uchli theorem, which is a key ingredient.

Our advance here stems from a careful analysis of the Halpern–L�uchli 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 key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken's tree theorem. This enables us to build solutions to the full Milliken's tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken's tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove 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. In particular, we 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.

� propos de l?auteur

Paul-Elliot Angles D'Auriac, Universite Claude Bernard Lyon 1, France.

Peter A. Cholak, University of Notre Dame, Indiana.

Damir D. Dzhafarov, University of Connecticut, Storrs, Connecticut.

Benoit Monin, Laboratoire d'Algorithmique, Complexite et Logique (LACL), Paris, France.

Ludovic Patey, Universite Claude Bernard Lyon 1, France.

Les informations fournies dans la section � A propos du livre � peuvent faire r�f�rence � une autre �dition de ce titre.

Description de la librairie

Visitez la page d�accueil du vendeur

Informations commerciales relatives au vendeur

RAREWAVES.COM LIMITED
Elsley Court, 20-22 Great Titchfield Street, London, W1W 8BE, United Kingdom

Conditions de vente et de livraison

Conditions g�n�rales de vente

Droit de r�tractation

Si vous �tes un consommateur, vous pouvez vous r�tracter du pr�sent contrat dans les conditions indiqu�es ci-dessous. Etre un consommateur signifie �tre une personne physique qui agit � des fins qui n'entrent pas dans le cadre de son activit� commerciale, industrielle, artisanale ou lib�rale.

Informations standardis�es sur la r�tractation

Droit de r�tractation:

Vous avez le droit d'annuler ce contrat dans les 14 jours pour quelque raison que ce soit.

Le d�lai de r�tractation expire quatorze jours apr�s le jour o� vous-m�me, ou un tiers autre que le transporteur et d�sign� par vous, prend physiquement possession du dernier bien, lot ou pi�ce.

Pour exercer le droit de r�tractation, vous devez nous Rarewaves.com UK, Unit 144 The Lightbox, 111 Power Road, W4 5PY, London, London, United Kingdom, notifier votre d�cision de r�tractation du pr�sent contrat au moyen d'une d�claration d�nu�e d'ambigu�t� (par exemple, lettre envoy�e par la poste, t�l�copie ou courrier �lectronique). Vous pouvez utiliser le mod�le de formulaire de r�tractation mais ce n'est pas obligatoire. Vous pouvez �galement remplir et transmettre une d�claration claire � cette fin sur notre site internet � �Vos commandes� dans �Votre compte�. Si vous utilisez cette option, nous vous enverrons sans d�lai un accus� de r�ception de la r�tractation sur un support durable (par exemple, par courriel).

Pour que le d�lai de r�tractation soit respect�, il suffit que vous transmettiez votre communication relative � l'exercice du droit de r�tractation avant l'expiration du d�lai de r�tractation.

Effets de la r�tractation :

En cas de r�tractation de votre part du pr�sent contrat, nous vous rembourserons tous les paiements re�us de vous, y compris les frais de livraison (� l'exception des frais suppl�mentaires d�coulant du fait que vous avez choisi, le cas �ch�ant, un mode de livraison autre que le mode moins co�teux de livraison standard propos� par nous) sans retard excessif et, en tout �tat de cause, au plus tard quatorze jours � compter du jour o� nous sommes inform�s de votre d�cision de r�tractation du pr�sent contrat. Nous proc�derons au remboursement en utilisant le m�me moyen de paiement que celui que vous aurez utilis� pour la transaction initiale, sauf si vous convenez express�ment d'un moyen diff�rent; en tout �tat de cause, ce remboursement n'occasionnera pas de frais pour vous.

Nous pouvons diff�rer le remboursement jusqu'� ce que nous ayons re�u le bien ou jusqu'� ce que vous ayez fourni une preuve d'exp�dition du bien, la date retenue �tant celle du premier de ces faits.

Vous devrez renvoyer ou rendre le bien, � Rarewaves.com UK, Unit 144 The Lightbox, 111 Power Road, W4 5PY, London, London, United Kingdom, sans retard excessif et, en tout �tat de cause, au plus tard quatorze jours apr�s que vous nous aurez communiqu� votre d�cision de r�tractation du pr�sent contrat. Ce d�lai est r�put� respect� si vous renvoyez le bien avant l'expiration du d�lai de quatorze jours. Vous devrez prendre en charge les frais directs de renvoi du bien. Votre responsabilit� n'est engag�e qu'� l'�gard de la d�pr�ciation du bien r�sultant de manipulations autres que celles n�cessaires pour �tablir la nature, les caract�ristiques et le bon fonctionnement de ce bien.

Exceptions au droit de r�tractation

Le droit de r�tractation ne s'applique pas � :

la fourniture d'un contenu num�rique non fourni sur un support mat�riel (tel qu'un CD ou un DVD), si vous avez accept� au moment de la passation de votre commande que nous commencions la fourniture de celui-ci et renonc� � votre droit de r�tractation ;
la fourniture d'un journal, d'un p�riodique ou d'un magazine, sauf pour les contrats d'abonnement � ces publications.

Formulaire de r�tractation

(A compl�ter et renvoyer seulement si vous souhaitez-vous r�tracter du pr�sent contrat)

A l'attention de : (Rarewaves.com UK, Unit 144 The Lightbox, 111 Power Road, W4 5PY, London, London, United Kingdom)

Je vous notifie par la pr�sente ma r�tractation du contrat portant sur la vente des produits suivants :

num�ro de commande :

Command� le/re�u le :

Mon nom et mon adresse :

Signature (seulement si ce formulaire est notifi� en version papier) :

Date

Conditions d'exp�dition

Please note that we do not offer Priority shipping to any country.

We currently do not ship to the below countries:

Russia
Belarus
Ukraine
Israel

Please do not attempt to place orders with any of these countries as a ship to address - they will be cancelled.

Frais d?exp�dition de Royaume-Uni�vers�Etats-Unis

Frais d?exp�dition de Royaume-Uni�vers�Etats-Unis
Quantit� command�e	60 � 60 jours ouvr�s	60 � 60 jours ouvr�s
Premier article	EUR 75.02	EUR 115.42

Les d�lais de livraison sont fix�s par les vendeurs et varient en fonction du transporteur et du lieu. Les commandes transitant par les douanes peuvent �tre retard�es et les acheteurs sont responsables de tous les droits ou frais associ�s. Les vendeurs peuvent vous contacter au sujet de frais suppl�mentaires afin de couvrir toute augmentation des co�ts d'exp�dition de vos articles.