Bulletin of the London Mathematical Society最新文献

{"title":"The Haagerup property is not invariant under quasi-isometry","authors":"Mathieu Carette,&nbsp;Kevin Whyte,&nbsp;Sylvain Arnt,&nbsp;Thibault Pillon,&nbsp;Alain Valette","doi":"10.1112/blms.70516","DOIUrl":"https://doi.org/10.1112/blms.70516","url":null,"abstract":"<p>Using the work of Cornulier–Valette and Whyte (see Appendix A for the latter), we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups. Appendix B shows, using the same examples, that the same holds for vanishing of the equivariant <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math>-compression.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 11","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70516","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177119","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":"A complex-analytic characterization of Lagrangian immersions in \u0000 \u0000 \u0000 C\u0000 n\u0000 \u0000 $mathbb {C}^n$\u0000 with transverse double points","authors":"Purvi Gupta,&nbsp;Rudranil Sahu","doi":"10.1112/blms.70511","DOIUrl":"https://doi.org/10.1112/blms.70511","url":null,"abstract":"&lt;p&gt;Given a compact smooth totally real immersed &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-submanifold &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Msubset mathbb {C}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with only finitely many transverse double points, it is known that if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is Lagrangian with respect to some Kähler form on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {C}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then it is rationally convex in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {C}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, but the converse is not true. We show that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is Lagrangian with respect to some Kähler form on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {C}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; if and only if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is rationally convex &lt;i&gt;and&lt;/i&gt; at each double point, the pair of transverse tangent planes to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfies the following diagonalizability condition: There is a complex linear transformation on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {C}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that maps the pair to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146295","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":"Pointwise character bounds for \u0000 \u0000 \u0000 SU\u0000 (\u0000 3\u0000 )\u0000 \u0000 $mathrm{SU}(3)$","authors":"Yunfeng Zhang","doi":"10.1112/blms.70519","DOIUrl":"https://doi.org/10.1112/blms.70519","url":null,"abstract":"<p>We present a basic pointwise bound for the irreducible characters of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>SU</mi>\u0000 <mo>(</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{SU}(3)$</annotation>\u0000 </semantics></math> and, as an application, derive new <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> bounds for these characters. Our approach is based on the descent of characters to singular sets and the cancellation in this formula.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146448","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":"Topological and geometric rigidity of nonnegatively curved submanifolds","authors":"Theodoros Vlachos","doi":"10.1112/blms.70524","DOIUrl":"https://doi.org/10.1112/blms.70524","url":null,"abstract":"<p>We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature satisfying a lower bound on the sectional curvature that depends only on the length of the mean curvature vector of the immersion. We show that this condition imposes strong constraints on either the topology or geometry of the submanifold. More precisely, we prove that, for dimensions <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation>$ngeqslant 5$</annotation>\u0000 </semantics></math>, the manifold is homeomorphic to a sphere, whereas in dimension four it is either diffeomorphic to <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <annotation>$mathbb {S}^4$</annotation>\u0000 </semantics></math> or isometric to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathord {mathbb {C}}mathord {P}^2$</annotation>\u0000 </semantics></math>, in which case the immersion is the standard embedding into <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>7</mn>\u0000 </msup>\u0000 <annotation>$mathbb {S}^7$</annotation>\u0000 </semantics></math> followed by an umbilical inclusion. Additionally, we provide examples showing that the spherical conclusion cannot in general be strengthened to an extrinsic rigidity statement, since there exist many geometrically distinct spheres satisfying the pinching condition.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70524","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146449","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 asymptotic Lebesgue's universal covering problem","authors":"A. Arman,&nbsp;A. Bondarenko,&nbsp;A. Prymak,&nbsp;D. Radchenko","doi":"10.1112/blms.70514","DOIUrl":"https://doi.org/10.1112/blms.70514","url":null,"abstract":"&lt;p&gt;A universal cover in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {E}^{n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$n=2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;). A simple universal cover in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {E}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is provided by the classical theorem of Jung, which states that any set of diameter 1 in an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional Euclidean space is contained in a ball &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$J_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of radius &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msqrt&gt;\u0000 &lt;mstyle&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;/mstyle&gt;\u0000 &lt;/msqrt&gt;\u0000 &lt;annotation&gt;$sqrt {tfrac{n}{2n+2}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;; in other words, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$J_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a universal cover in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {E}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We show that in high dimensions, Jung's ball &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70514","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146014","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":"Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur","authors":"Marek Cúth,&nbsp;Ondřej F. K. Kalenda","doi":"10.1112/blms.70518","DOIUrl":"https://doi.org/10.1112/blms.70518","url":null,"abstract":"<p>We prove that the Lipschitz-free space over any uniformly discrete metric space has the 3-Schur property.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70518","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145567","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":"The sharp threshold for rainbow stackings of random edge-colourings","authors":"Hong Liu,&nbsp;Guorui Ma,&nbsp;Yangrui Xiang,&nbsp;Zhifei Yan","doi":"10.1112/blms.70522","DOIUrl":"https://doi.org/10.1112/blms.70522","url":null,"abstract":"<p>A rainbow stacking of <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math> independent, uniformly random <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-edge-colourings of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$K_n$</annotation>\u0000 </semantics></math> is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette size <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math> required for the existence of such stackings was recently initiated by Alon, Defant and Kravitz (<i>Bull. Lond. Math. Soc</i>. 57 (2025), 1656–1670), who bounded the phase transition within a constant-order window around <span></span><math>\u0000 <semantics>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>log</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>!</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mfrac>\u0000 <annotation>$frac{mbinom{n}{2}}{2log (n!)}$</annotation>\u0000 </semantics></math>. We determine the constant term in this transition. For every fixed <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$mgeqslant 2$</annotation>\u0000 </semantics></math> and every function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ω</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$omega (n)rightarrow infty$</annotation>\u0000 </semantics></math>, with high probability there is no rainbow stacking if\u0000\u0000 </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146017","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":"No-\u0000 \u0000 \u0000 (\u0000 k\u0000 +\u0000 1\u0000 )\u0000 \u0000 $(k+1)$\u0000 -in-line problem for large constant \u0000 \u0000 k\u0000 $k$","authors":"Alexandr Grebennikov,&nbsp;Matthew Kwan","doi":"10.1112/blms.70510","DOIUrl":"https://doi.org/10.1112/blms.70510","url":null,"abstract":"<p>How many points can be placed in an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>×</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$ntimes n$</annotation>\u0000 </semantics></math> grid so that every (affine) line contains at most <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> points? We prove that for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mi>k</mi>\u0000 <mo>⩾</mo>\u0000 <msup>\u0000 <mn>10</mn>\u0000 <mn>37</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$n geqslant k geqslant 10^{37}$</annotation>\u0000 </semantics></math> the maximum number of points is exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$kn$</annotation>\u0000 </semantics></math>. Our proof builds on the recent work of Kovács, Nagy and Szabó (who proved an analogous result when <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> is at least about <span></span><math>\u0000 <semantics>\u0000 <msqrt>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mi>log</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msqrt>\u0000 <annotation>$sqrt {n log n}$</annotation>\u0000 </semantics></math>), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher-dimensional extensions of this problem.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70510","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145851","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":"Equivalence between Erdős–Hajnal and polynomial Rödl and Nikiforov conjectures","authors":"Matija Bucić,&nbsp;Jacob Fox,&nbsp;Huy Tuan Pham","doi":"10.1112/blms.70493","DOIUrl":"https://doi.org/10.1112/blms.70493","url":null,"abstract":"<p>It is well-known that polynomial versions of theorems of Rödl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erdős–Hajnal conjecture. In this note, we prove that these three conjectures are in fact equivalent, extending several previous particular results in this direction by Fox, Nguyen, Scott, and Seymour; Nguyen, Scott, and Seymour; and Gishboliner and Shapira. We deduce that the family of string graphs satisfies the polynomial Rödl conjecture. We also derive analogous results for hypergraphs, tournaments, ordered graphs, and colored graphs.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70493","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145849","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":"Canonical metrics on holomorphic fibre bundles","authors":"John Benjamin McCarthy","doi":"10.1112/blms.70509","DOIUrl":"https://doi.org/10.1112/blms.70509","url":null,"abstract":"<p>In this article, we completely describe the existence of canonical metrics, known as optimal symplectic connections, on isotrivial Kähler fibrations. In this setting an optimal symplectic connection is induced from a Hermite–Einstein connection on the holomorphic principal bundle of relative automorphisms, and the Hitchin–Kobayashi correspondence asserts the existence of such a connection precisely when the principal bundle is polystable. Combined with results of Dervan and Sektnan this generates many new examples of constant scalar curvature Kähler (cscK) metrics on the total space of holomorphic fibre bundles. Our results indicate that in general the optimal symplectic connection equation should be viewed as a generalisation of the Hermite–Einstein equation to holomorphic fibrations where the complex structure of the fibres varies.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70509","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145852","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}