Pub Date : 2019-04-23DOI: 10.1142/s021906132150015x
Shlomo Eshel, Itay Kaplan
Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.
{"title":"On uniform definability of types over finite sets for NIP formulas","authors":"Shlomo Eshel, Itay Kaplan","doi":"10.1142/s021906132150015x","DOIUrl":"https://doi.org/10.1142/s021906132150015x","url":null,"abstract":"Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"43 1","pages":"2150015:1-2150015:13"},"PeriodicalIF":0.9,"publicationDate":"2019-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87288760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-23DOI: 10.1142/S0219061321500276
Thomas Gilton, I. Neeman
Author(s): Gilton, Thomas; Neeman, Itay | Abstract: We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{aleph_0}=aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a Partition Product, and thereby obtain a model of this axiom in which $2^{aleph_0}=aleph_3$.
{"title":"Abraham-Rubin-Shelah open colorings and a large continuum","authors":"Thomas Gilton, I. Neeman","doi":"10.1142/S0219061321500276","DOIUrl":"https://doi.org/10.1142/S0219061321500276","url":null,"abstract":"Author(s): Gilton, Thomas; Neeman, Itay | Abstract: We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{aleph_0}=aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a Partition Product, and thereby obtain a model of this axiom in which $2^{aleph_0}=aleph_3$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"56 1","pages":"2150027:1-2150027:55"},"PeriodicalIF":0.9,"publicationDate":"2019-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82272482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-04DOI: 10.1142/S0219061321500185
M. Goldstern, Jakob Kellner, D. Mej'ia, S. Shelah
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber.
{"title":"Controlling cardinal characteristics without adding reals","authors":"M. Goldstern, Jakob Kellner, D. Mej'ia, S. Shelah","doi":"10.1142/S0219061321500185","DOIUrl":"https://doi.org/10.1142/S0219061321500185","url":null,"abstract":"We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"1134 1","pages":"2150018:1-2150018:29"},"PeriodicalIF":0.9,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79395227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-20DOI: 10.1142/s021906132150029x
A. Nies, Philipp Schlicht, K. Tent
Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup [Formula: see text] of [Formula: see text] is called oligomorphic if for each [Formula: see text], its natural action on [Formula: see text]-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of [Formula: see text], and a ternary relation [Formula: see text]. This structure derived from [Formula: see text] was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3]. If [Formula: see text] has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of [Formula: see text] from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.
令[公式:见文]表示自然数排列的拓扑群。如果对于每个[公式:见文]，其对[公式:见文]-自然数元组的自然作用只有有限多个轨道，则[公式:见文]的封闭子群[公式:见文]被称为寡胚。在波兰空间上等价关系之间Borel约化的情况下，研究了[公式:见文]的低纯子群上拓扑同构关系的复杂性。给定[公式:见文]的一个闭子群[公式:见文]，粗群[公式:见文]是具有[公式:见文]的开子群的余集域和三元关系[公式:见文]的结构。这个结构来源于[公式:见文]，在[A]中被引入。Kechris, A. Nies和K. Tent，拓扑群同构的复杂性[j].符号逻辑83(3)(2018):190 - 1203,Sec. 3.3。如果[Formula: see text]只有可数的开放子群，则[Formula: see text]是一个可数结构。粗群是我们研究此类封闭子群的主要工具[公式:见原文]。我们将它们抽象地公理化为具有三元关系的结构。对于低纯群和无限群，我们在群和相应的粗群之间建立了stone型对偶。特别地，我们可以用Borel的方式从[Formula: see text]的粗群中恢复一个同构副本。我们利用这个对偶证明了[公式:见文]的低纯子群的同构关系是Borel可约为所有类可数的Borel等价关系。我们证明了相同的上界适用于[公式:见文本]中拓扑同构于寡纯群的更大的闭子群类。
{"title":"Coarse groups, and the isomorphism problem for oligomorphic groups","authors":"A. Nies, Philipp Schlicht, K. Tent","doi":"10.1142/s021906132150029x","DOIUrl":"https://doi.org/10.1142/s021906132150029x","url":null,"abstract":"Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup [Formula: see text] of [Formula: see text] is called oligomorphic if for each [Formula: see text], its natural action on [Formula: see text]-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of [Formula: see text], and a ternary relation [Formula: see text]. This structure derived from [Formula: see text] was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3]. If [Formula: see text] has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of [Formula: see text] from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"PP 1","pages":"2150029:1-2150029:31"},"PeriodicalIF":0.9,"publicationDate":"2019-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84860043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-18DOI: 10.1142/S0219061320500014
F. Calderoni, H. Mildenberger, L. Ros
Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].
{"title":"Uncountable structures are not classifiable up to bi-embeddability","authors":"F. Calderoni, H. Mildenberger, L. Ros","doi":"10.1142/S0219061320500014","DOIUrl":"https://doi.org/10.1142/S0219061320500014","url":null,"abstract":"Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"28 1","pages":"2050001"},"PeriodicalIF":0.9,"publicationDate":"2019-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79202330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-26DOI: 10.1142/s0219061320500245
Will Boney, M. Lieberman
We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].
{"title":"Tameness, powerful images, and large cardinals","authors":"Will Boney, M. Lieberman","doi":"10.1142/s0219061320500245","DOIUrl":"https://doi.org/10.1142/s0219061320500245","url":null,"abstract":"We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"18 1","pages":"2050024:1-2050024:18"},"PeriodicalIF":0.9,"publicationDate":"2019-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81536829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-15DOI: 10.1142/S0219061319500132
Sandra Müller, R. Schindler, W. Woodin
We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy.
{"title":"Mice with finitely many Woodin cardinals from optimal determinacy hypotheses","authors":"Sandra Müller, R. Schindler, W. Woodin","doi":"10.1142/S0219061319500132","DOIUrl":"https://doi.org/10.1142/S0219061319500132","url":null,"abstract":"We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"28 1","pages":"1950013:1-1950013:118"},"PeriodicalIF":0.9,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83237675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-31DOI: 10.1142/S0219061321500112
F. Wehrung
Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L ∞λ. We prove that many naturally defined classes are anti-elementary, including the following: • the class of all lattices of finitely generated convex l-subgroups of members of any class of l-groups containing all Archimedean l-groups; • the class of all semilattices of finitely generated l-ideals of members of any nontrivial quasivariety of l-groups; • the class of all Stone duals of spectra of MV-algebras-this yields a negative solution for the MV-spectrum Problem; • the class of all semilattices of finitely generated two-sided ideals of rings; • the class of all semilattices of finitely generated submodules of modules; • the class of all monoids encoding the nonstable K_0-theory of von Neumann regular rings, respectively C*-algebras of real rank zero; • (assuming arbitrarily large Erd˝os cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4-frame. The main underlying principle is that under quite general conditions, for a functor Φ : A → B, if there exists a non-commutative diagram D of A, indexed by a common sort of poset called an almost join-semilattice, such that • Φ D^I is a commutative diagram for every set I, • Φ D is not isomorphic to Φ X for any commutative diagram X in A, then the range of Φ is anti-elementary.
{"title":"From noncommutative diagrams to anti-elementary classes","authors":"F. Wehrung","doi":"10.1142/S0219061321500112","DOIUrl":"https://doi.org/10.1142/S0219061321500112","url":null,"abstract":"Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L ∞λ. We prove that many naturally defined classes are anti-elementary, including the following: • the class of all lattices of finitely generated convex l-subgroups of members of any class of l-groups containing all Archimedean l-groups; • the class of all semilattices of finitely generated l-ideals of members of any nontrivial quasivariety of l-groups; • the class of all Stone duals of spectra of MV-algebras-this yields a negative solution for the MV-spectrum Problem; • the class of all semilattices of finitely generated two-sided ideals of rings; • the class of all semilattices of finitely generated submodules of modules; • the class of all monoids encoding the nonstable K_0-theory of von Neumann regular rings, respectively C*-algebras of real rank zero; • (assuming arbitrarily large Erd˝os cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4-frame. The main underlying principle is that under quite general conditions, for a functor Φ : A → B, if there exists a non-commutative diagram D of A, indexed by a common sort of poset called an almost join-semilattice, such that • Φ D^I is a commutative diagram for every set I, • Φ D is not isomorphic to Φ X for any commutative diagram X in A, then the range of Φ is anti-elementary.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"45 1","pages":"2150011:1-2150011:56"},"PeriodicalIF":0.9,"publicationDate":"2019-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77701518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-20DOI: 10.1142/s0219061322500180
Natasha Dobrinen
For $kge 3$, the Henson graph $mathcal{H}_k$ is the analogue of the Rado graph in which $k$-cliques are forbidden. Building on the author's result for $mathcal{H}_3$, we prove that for each $kge 4$, $mathcal{H}_k$ has finite big Ramsey degrees: To each finite $k$-clique-free graph $G$, there corresponds an integer $T(G,mathcal{H}_k)$ such that for any coloring of the copies of $G$ in $mathcal{H}_k$ into finitely many colors, there is a subgraph of $mathcal{H}_k$, again isomorphic to $mathcal{H}_k$, in which the coloring takes no more than $T(G, mathcal{H}_k)$ colors. Prior to this article, the Ramsey theory of $mathcal{H}_k$ for $kge 4$ had only been resolved for vertex colorings by El-Zahar and Sauer in 1989. We develop a unified framework for coding copies of $mathcal{H}_k$ into a new class of trees, called strong $mathcal{H}_k$-coding trees, and prove Ramsey theorems for these trees, forming a family of Halpern-Lauchli and Milliken-style theorems which are applied to deduce finite big Ramsey degrees. The approach here streamlines the one in cite{DobrinenH_317} for $mathcal{H}_3$ and provides a general methodology opening further study of big Ramsey degrees for homogeneous structures with forbidden configurations. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and recent work of Zucker.
{"title":"The Ramsey theory of Henson graphs","authors":"Natasha Dobrinen","doi":"10.1142/s0219061322500180","DOIUrl":"https://doi.org/10.1142/s0219061322500180","url":null,"abstract":"For $kge 3$, the Henson graph $mathcal{H}_k$ is the analogue of the Rado graph in which $k$-cliques are forbidden. Building on the author's result for $mathcal{H}_3$, we prove that for each $kge 4$, $mathcal{H}_k$ has finite big Ramsey degrees: To each finite $k$-clique-free graph $G$, there corresponds an integer $T(G,mathcal{H}_k)$ such that for any coloring of the copies of $G$ in $mathcal{H}_k$ into finitely many colors, there is a subgraph of $mathcal{H}_k$, again isomorphic to $mathcal{H}_k$, in which the coloring takes no more than $T(G, mathcal{H}_k)$ colors. Prior to this article, the Ramsey theory of $mathcal{H}_k$ for $kge 4$ had only been resolved for vertex colorings by El-Zahar and Sauer in 1989. We develop a unified framework for coding copies of $mathcal{H}_k$ into a new class of trees, called strong $mathcal{H}_k$-coding trees, and prove Ramsey theorems for these trees, forming a family of Halpern-Lauchli and Milliken-style theorems which are applied to deduce finite big Ramsey degrees. The approach here streamlines the one in cite{DobrinenH_317} for $mathcal{H}_3$ and provides a general methodology opening further study of big Ramsey degrees for homogeneous structures with forbidden configurations. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and recent work of Zucker.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"162 1","pages":"2250018:1-2250018:88"},"PeriodicalIF":0.9,"publicationDate":"2019-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83856817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-17DOI: 10.1142/s0219061320500087
Yatir Halevi, Assaf Hasson, Franziska Jahnke
We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.
我们开始研究可定义的[公式:见文]-拓扑，并证明在[公式:见文]-henselian NIP域上最多有一个这样的[公式:见文]-拓扑。同样地，我们证明，如果[Formula: see text]是一个双值NIP字段，具有[Formula: see text] henselian(分别为[Formula: see text]-henselian)，则[Formula: see text]和[Formula: see text]具有可比性(分别为依赖性)。因此，Shelah的NIP域猜想暗示了NIP域的henselianity猜想。进一步证明了后一个猜想对于任何具有最小残差域的域都具有henselian值。我们通过证明Shelah的猜想等价于任何不包含在有限域的代数闭包中的NIP域都是-henselian的命题来得出结论。
{"title":"Definable V-topologies, Henselianity and NIP","authors":"Yatir Halevi, Assaf Hasson, Franziska Jahnke","doi":"10.1142/s0219061320500087","DOIUrl":"https://doi.org/10.1142/s0219061320500087","url":null,"abstract":"We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"115 1","pages":"2050008:1-2050008:33"},"PeriodicalIF":0.9,"publicationDate":"2019-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86061973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：