Journal of the London Mathematical Society-Second Series最新文献

{"title":"On the annihilator variety of a highest weight module for classical Lie algebras","authors":"Zhanqiang Bai, Jia-Jun Ma, Yutong Wang","doi":"10.1112/jlms.70256","DOIUrl":"https://doi.org/10.1112/jlms.70256","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$mathfrak {g}$</annotation>\u0000 </semantics></math> be a classical complex simple Lie algebra, and let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>λ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$L(lambda)$</annotation>\u0000 </semantics></math> be the irreducible highest weight module of <span></span><math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$mathfrak {g}$</annotation>\u0000 </semantics></math> with the highest weight <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>−</mo>\u0000 <mi>ρ</mi>\u0000 </mrow>\u0000 <annotation>$lambda -rho$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>ρ</mi>\u0000 <annotation>$rho$</annotation>\u0000 </semantics></math> is half the sum of positive roots. The associated variety of the annihilator ideal of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>λ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$L(lambda)$</annotation>\u0000 </semantics></math> is known as the annihilator variety of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>λ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$L(lambda)$</annotation>\u0000 </semantics></math>. It is established by Joseph that the annihilator variety of a highest weight module is the Zariski closure of a nilpotent orbit in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$mathfrak {g}^*$</annotation>\u0000 </semantics></math>. However, describing this nilpotent orbit for a given highest weight module <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>λ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$L(lambda)$</annotation>\u0000 </semantics></math> can be quite challenging. In this paper, we present some efficient algorithms based on the Robinson–Schensted insertion algorithm to compute these orbits for classical Lie algebras. Our formulae are given by introducing two algorithms, that is, bipartition algorithm and partition algorithm. To get a special or metaplectic special partition from a domino type partition, we define the H-algorithm based on the Robinso","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144773747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"K-stable Fano threefolds of rank 2 and degree 28","authors":"Joseph Malbon","doi":"10.1112/jlms.70259","DOIUrl":"https://doi.org/10.1112/jlms.70259","url":null,"abstract":"<p>Moduli spaces of Fano varieties have historically been difficult to construct. However, recent work has shown that smooth K-polystable Fano varieties of fixed dimension and volume can be parametrised by a quasi-projective moduli space. In this paper, we prove that all smooth Fano threefolds with Picard rank 2 and degree 28 are K-polystable, except for some explicit cases which we describe.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70259","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144773685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arboreal Galois groups of postcritically finite quadratic polynomials: The periodic case","authors":"Robert L. Benedetto,&nbsp;Dragos Ghioca,&nbsp;Jamie Juul,&nbsp;Thomas J. Tucker","doi":"10.1112/jlms.70257","DOIUrl":"https://doi.org/10.1112/jlms.70257","url":null,"abstract":"<p>We provide an explicit construction of the arboreal Galois group for the postcritically finite polynomial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$f(z) = z^2 +c$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>c</mi>\u0000 <annotation>$c$</annotation>\u0000 </semantics></math> belongs to some arbitrary field of characteristic not equal to 2. In this first of two papers, we consider the case that the critical point is periodic.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Full capacity–volumetry of sharp exp-integrability law","authors":"David R. Adams,&nbsp;Jie Xiao","doi":"10.1112/jlms.70255","DOIUrl":"https://doi.org/10.1112/jlms.70255","url":null,"abstract":"<p>This paper uses law of trichotomy to show a full range of capacity–volumetry of the sharp <span></span><math>\u0000 <semantics>\u0000 <mi>exp</mi>\u0000 <annotation>$exp$</annotation>\u0000 </semantics></math>-integrability law which covers the sharp Adams–Moser–Trudinger <span></span><math>\u0000 <semantics>\u0000 <mi>exp</mi>\u0000 <annotation>$exp$</annotation>\u0000 </semantics></math>-integrability law for higher order derivatives, thereby finding a new approach to a relatively complete family of the essential capacity–volumetric estimates with the optimal constants including the sharp Ahlfors–Beurling–Pólya–Szegö and Morrey–Sobolev capacity–volumetric inequalities.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quasiregular mappings between equiregular sub-Riemannian manifolds","authors":"Chang-Yu Guo,&nbsp;Sebastiano Nicolussi Golo,&nbsp;Marshall Williams,&nbsp;Yi Xuan","doi":"10.1112/jlms.70254","DOIUrl":"https://doi.org/10.1112/jlms.70254","url":null,"abstract":"<p>In this paper, we provide an alternative approach to an expectation of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular sub-Riemannian manifolds of homogeneous dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$Qgeqslant 2$</annotation>\u0000 </semantics></math> has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis–Mostow, Karmanova, and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam, and Tyson to establish the necessary analytic foundations.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144740401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The second moment of the Riemann zeta function at its local extrema","authors":"Christopher Hughes,&nbsp;Solomon Lugmayer,&nbsp;Andrew Pearce-Crump","doi":"10.1112/jlms.70250","DOIUrl":"https://doi.org/10.1112/jlms.70250","url":null,"abstract":"<p>Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mrow>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>−</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>π</mi>\u0000 </mrow>\u0000 </mfrac>\u0000 <mi>T</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>log</mi>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$frac{e^2 - 5}{2 pi } T (log T)^2$</annotation>\u0000 </semantics></math>. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144725530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuum limit of fourth-order Schrödinger equations on the lattice","authors":"Jiawei Cheng,&nbsp;Bobo Hua","doi":"10.1112/jlms.70247","DOIUrl":"https://doi.org/10.1112/jlms.70247","url":null,"abstract":"<p>In this paper, we consider the discrete fourth-order Schrödinger equation on the lattice <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$hmathbb {Z}^2$</annotation>\u0000 </semantics></math>. Uniform Strichartz estimates are established by analyzing frequency localized oscillatory integrals with the method of stationary phase and applying Littlewood–Paley inequalities. As an application, we obtain the precise rate of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^2$</annotation>\u0000 </semantics></math> convergence from the solutions of discrete semilinear equations to those of the corresponding equations on the Euclidean plane <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math> in the continuum limit <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$h rightarrow 0$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory","authors":"David Fernández-Bretón,&nbsp;Jareb Navarro-Castillo,&nbsp;Jesús A. Soria-Rojas","doi":"10.1112/jlms.70249","DOIUrl":"https://doi.org/10.1112/jlms.70249","url":null,"abstract":"&lt;p&gt;We study ultrafilters from the perspective of the algebra in the Čech–Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a Q-point (resp., a selective ultrafilter) and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal F^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (resp., &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal G^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) is the smallest family containing &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and closed under iterated sums (resp., closed under Blass–Frolík sums and Rudin–Keisler images), then &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal F^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (resp., &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal G^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathnormal {mathbf {L}(mathbb {R})}[p]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathnormal {mathbf {L}(mathbb {R})}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a Solovay model (of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathnormal {mathsf {ZF}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; without choice) and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a selective ultrafilter, there are no idempotent elements. In particular, the theory &lt;sp","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A topological algorithm for the Fourier transform of Stokes data at infinity","authors":"Jean Douçot,&nbsp;Andreas Hohl","doi":"10.1112/jlms.70253","DOIUrl":"https://doi.org/10.1112/jlms.70253","url":null,"abstract":"<p>We give a topological description of the behaviour of Stokes matrices under the Fourier transform from infinity to infinity in a large number of cases of one level. This explicit, algorithmic statement is obtained by building on a recent result of T. Mochizuki about the Fourier transform of Stokes data of irregular connections on the Riemann sphere and by using the language of Stokes local systems due to P. Boalch. In particular, this induces explicit isomorphisms between wild character varieties, in a much larger range of examples than those for which such isomorphisms have previously been written down. We conjecture that these isomorphisms are compatible with the quasi-Hamiltonian structure on the wild character varieties.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Canonical colourings in random graphs","authors":"Nina Kamčev,&nbsp;Mathias Schacht","doi":"10.1112/jlms.70239","DOIUrl":"https://doi.org/10.1112/jlms.70239","url":null,"abstract":"&lt;p&gt;Rödl and Ruciński (&lt;i&gt;J. Amer. Math. Soc&lt;/i&gt;. &lt;b&gt;8&lt;/b&gt; (1995), 917–942) established Ramsey's theorem for random graphs. In particular, for fixed integers &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;annotation&gt;$r$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&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;$ell geqslant 2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; they proved that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;̂&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$hat{p}_{K_ell,r}(n)=n^{-frac{2}{ell +1}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a threshold for the Ramsey property that every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;annotation&gt;$r$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-colouring of the edges of the binomial random graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$G(n,p)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; yields a monochromatic copy of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$K_ell$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We investigate how this result extends to arbitr","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144705681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}