Mathematical Logic Quarterly最新文献

{"title":"Strongly unfoldable, splitting and bounding","authors":"Ömer Faruk Bağ,&nbsp;Vera Fischer","doi":"10.1002/malq.202200003","DOIUrl":"https://doi.org/10.1002/malq.202200003","url":null,"abstract":"<p>Assuming <math>\u0000 <semantics>\u0000 <mi>GCH</mi>\u0000 <annotation>$mathsf {GCH}$</annotation>\u0000 </semantics></math>, we show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under a one step iteration of the Hechler forcing for adding a dominating κ-real. Moreover, we show that if κ is strongly unfoldable, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$2^kappa =kappa ^+$</annotation>\u0000 </semantics></math> and λ is a regular cardinal such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation>$kappa ^+ &lt; lambda$</annotation>\u0000 </semantics></math>, then there is a set generic extension in which <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>b</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation>$mathfrak {s}(kappa ) = kappa ^+ &lt; mathfrak {b}(kappa ) = mathfrak {c}(kappa ) = lambda$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"7-14"},"PeriodicalIF":0.3,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50154033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on \u0000 \u0000 fsg\u0000 $text{fsg}$\u0000 groups in p-adically closed fields","authors":"Will Johnson","doi":"10.1002/malq.202200026","DOIUrl":"https://doi.org/10.1002/malq.202200026","url":null,"abstract":"<p>Let <i>G</i> be a definable group in a <i>p</i>-adically closed field <i>M</i>. We show that <i>G</i> has finitely satisfiable generics (<math>\u0000 <semantics>\u0000 <mtext>fsg</mtext>\u0000 <annotation>$text{fsg}$</annotation>\u0000 </semantics></math>) if and only if <i>G</i> is definably compact. The case <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$M = mathbb {Q}_p$</annotation>\u0000 </semantics></math> was previously proved by Onshuus and Pillay.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"50-57"},"PeriodicalIF":0.3,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50154036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Decomposition into special submanifolds","authors":"Masato Fujita","doi":"10.1002/malq.202200057","DOIUrl":"https://doi.org/10.1002/malq.202200057","url":null,"abstract":"<p>We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"104-116"},"PeriodicalIF":0.3,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The power set and the set of permutations with finitely many non-fixed points of a set","authors":"Guozhen Shen","doi":"10.1002/malq.202100070","DOIUrl":"https://doi.org/10.1002/malq.202100070","url":null,"abstract":"&lt;p&gt;For a cardinal &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {a}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we write &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mtext&gt;fin&lt;/mtext&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for the cardinality of the set of permutations with finitely many non-fixed points of a set which is of cardinality &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {a}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We investigate the relationships between &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$2^mathfrak {a}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mtext&gt;fin&lt;/mtext&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for an arbitrary infinite cardinal &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {a}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathsf {ZF}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (without the axiom of choice). It is proved in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathsf {ZF}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;≠&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mtext&gt;fin&lt;/mtext&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$2^mathfrak {a}ne operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for all inf","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"40-45"},"PeriodicalIF":0.3,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Contents: (Math. Log. Quart. 4/2022)","authors":"","doi":"10.1002/malq.202240000","DOIUrl":"10.1002/malq.202240000","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"391-393"},"PeriodicalIF":0.3,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202240000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80624922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Piece selection and cardinal arithmetic","authors":"Pierre Matet","doi":"10.1002/malq.202100033","DOIUrl":"10.1002/malq.202100033","url":null,"abstract":"<p>We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>≥</mo>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$lambda ge 2^kappa$</annotation>\u0000 </semantics></math>, then (a) <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$I_{kappa , lambda }$</annotation>\u0000 </semantics></math> is not (λ, 2)-distributive, and (b) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>→</mo>\u0000 <msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$I_{kappa , lambda }^+ rightarrow (I_{kappa , lambda }^+)^2_omega$</annotation>\u0000 </semantics></math> does not hold.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"416-446"},"PeriodicalIF":0.3,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74802387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Intuitionistic propositional probability logic","authors":"Anelina Ilić-Stepić,&nbsp;Mateja Knežević,&nbsp;Zoran Ognjanović","doi":"10.1002/malq.202100052","DOIUrl":"10.1002/malq.202100052","url":null,"abstract":"<p>We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and an intuitionistic tableaux method.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"479-495"},"PeriodicalIF":0.3,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83342708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Automorphism invariant measures and weakly generic automorphisms","authors":"Gábor Sági","doi":"10.1002/malq.202100044","DOIUrl":"10.1002/malq.202100044","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a countable ℵ&lt;sub&gt;0&lt;/sub&gt;-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;; the secondary motivation is to study the existence of weakly generic automorphisms of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of &lt;i&gt;A&lt;/i&gt; and of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;; we also present sufficient conditions implying that the theory of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is amenable. More concretely, we show that if the set of locally finite automorphisms of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is dense (in particular, if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure μ on the subsets of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; definable with parameters such that μ is invariant under &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Moreover, if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"458-478"},"PeriodicalIF":0.3,"publicationDate":"2022-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85168891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tameness of definably complete locally o-minimal structures and definable bounded multiplication","authors":"Masato Fujita,&nbsp;Tomohiro Kawakami,&nbsp;Wataru Komine","doi":"10.1002/malq.202200004","DOIUrl":"10.1002/malq.202200004","url":null,"abstract":"<p>We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"496-515"},"PeriodicalIF":0.3,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89499923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on edge colorings and trees","authors":"Adi Jarden,&nbsp;Ziv Shami","doi":"10.1002/malq.202100019","DOIUrl":"10.1002/malq.202100019","url":null,"abstract":"<p>We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>μ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$mu ^+&lt;kappa$</annotation>\u0000 </semantics></math>. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property, and for each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>,</mo>\u0000 <mi>μ</mi>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$lambda ,mu &lt;kappa$</annotation>\u0000 </semantics></math> there exists <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$kappa ^*&lt;kappa$</annotation>\u0000 </semantics></math> such that every tree of height μ with λ nodes has less than <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$kappa ^*$</annotation>\u0000 </semantics></math> branches.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"447-457"},"PeriodicalIF":0.3,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75494728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}