{"title":"The Turán number of surfaces","authors":"Maya Sankar","doi":"10.1112/blms.13167","DOIUrl":"https://doi.org/10.1112/blms.13167","url":null,"abstract":"<p>We show that there is a constant <span></span><math>\u0000 <semantics>\u0000 <mi>c</mi>\u0000 <annotation>$c$</annotation>\u0000 </semantics></math> such that any 3-uniform hypergraph <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>${mathcal {H}}$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> vertices and at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <msup>\u0000 <mi>n</mi>\u0000 <mrow>\u0000 <mn>5</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$cn^{5/2}$</annotation>\u0000 </semantics></math> edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3786-3800"},"PeriodicalIF":0.8,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142851521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Deconstructible abstract elementary classes of modules and categoricity","authors":"Jan Šaroch, Jan Trlifaj","doi":"10.1112/blms.13172","DOIUrl":"https://doi.org/10.1112/blms.13172","url":null,"abstract":"<p>We prove a version of Shelah's categoricity conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> is a deconstructible class of modules that fits in an abstract elementary class <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>,</mo>\u0000 <mo>⪯</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mathcal {A},preceq)$</annotation>\u0000 </semantics></math> such that (1) <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> is closed under direct summands and (2) <span></span><math>\u0000 <semantics>\u0000 <mo>⪯</mo>\u0000 <annotation>$preceq$</annotation>\u0000 </semantics></math> refines direct summands, then <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> is closed under arbitrary direct limits. In the Appendix, we prove that the assumption (2) is not needed in some models of ZFC.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3854-3866"},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13172","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximation properties of torsion classes","authors":"Sean Cox, Alejandro Poveda, Jan Trlifaj","doi":"10.1112/blms.13169","DOIUrl":"https://doi.org/10.1112/blms.13169","url":null,"abstract":"<p>We strengthen a result of Bagaria and Magidor (<i>Trans. Amer. Math. Soc</i>. <b>366</b> (2014), no. 4, 1857–1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove that\u0000\u0000 </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3819-3828"},"PeriodicalIF":0.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13169","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142851443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Turán densities for daisies and hypercubes","authors":"David Ellis, Maria-Romina Ivan, Imre Leader","doi":"10.1112/blms.13171","DOIUrl":"https://doi.org/10.1112/blms.13171","url":null,"abstract":"<p>An <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-daisy is an <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-uniform hypergraph consisting of the six <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-sets formed by taking the union of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>r</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(r-2)$</annotation>\u0000 </semantics></math>-set with each of the 2-sets of a disjoint 4-set. Bollobás, Leader and Malvenuto, and also Bukh, conjectured that the Turán density of the <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-daisy tends to zero as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$r rightarrow infty$</annotation>\u0000 </semantics></math>. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Turán densities of hypercubes. For fixed <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> and large <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>, we show that the smallest set of vertices of the <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-dimensional hypercube <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$Q_n$</annotation>\u0000 </semantics></math> that intersects every copy of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>d</mi>\u0000 </msub>\u0000 <annotation>$Q_d$</annotation>\u0000 </semantics></math> has asymptotic density strictly below <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3838-3853"},"PeriodicalIF":0.8,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13171","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Herglotz's representation and Carathéodory's approximation","authors":"Tirthankar Bhattacharyya, Mainak Bhowmik, Poornendu Kumar","doi":"10.1112/blms.13165","DOIUrl":"https://doi.org/10.1112/blms.13165","url":null,"abstract":"<p>Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain <span></span><math>\u0000 <semantics>\u0000 <mi>Ω</mi>\u0000 <annotation>$Omega$</annotation>\u0000 </semantics></math>, a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator-valued functions on a multi-connected domain.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3752-3776"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Segre products of cluster algebras","authors":"Jan E. Grabowski, Lauren Hindmarch","doi":"10.1112/blms.13166","DOIUrl":"https://doi.org/10.1112/blms.13166","url":null,"abstract":"<p>We show that under mild assumptions the Segre product of two graded cluster algebras has a natural cluster algebra structure.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3777-3785"},"PeriodicalIF":0.8,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142851489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On subgroups with narrow Schreier graphs","authors":"Pénélope Azuelos","doi":"10.1112/blms.13157","DOIUrl":"https://doi.org/10.1112/blms.13157","url":null,"abstract":"<p>We study finitely generated pairs of groups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 <mo>⩽</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$H leqslant G$</annotation>\u0000 </semantics></math> such that the Schreier graph of <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> has at least two ends and is <i>narrow</i>. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> is a virtual fiber subgroup if and only if <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> contains infinitely many double cosets of <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math>. Along the way, we prove that if a group acts essentially on a finite-dimensional CAT(0) cube complex with no facing triples, then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3652-3668"},"PeriodicalIF":0.8,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142851437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Reflecting perfection for finite-dimensional differential graded algebras","authors":"Isambard Goodbody","doi":"10.1112/blms.13160","DOIUrl":"https://doi.org/10.1112/blms.13160","url":null,"abstract":"<p>We generalise two facts about finite-dimensional algebras to finite-dimensional differential graded algebras. The first is the Nakayama lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality, respectively. As an application, we prove a corepresentability result for finite-dimensional differential graded algebras.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3689-3707"},"PeriodicalIF":0.8,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Birational geometry of Beauville–Mukai systems III: Asymptotic behavior","authors":"Xuqiang Qin, Justin Sawon","doi":"10.1112/blms.13158","DOIUrl":"https://doi.org/10.1112/blms.13158","url":null,"abstract":"<p>Suppose that a Hilbert scheme of points on a K3 surface <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3669-3680"},"PeriodicalIF":0.8,"publicationDate":"2024-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142862331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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