Combinatorica最新文献

{"title":"Connected Fair Detachments of Hypergraphs I","authors":"Amin Bahmanian","doi":"10.1007/s00493-026-00224-z","DOIUrl":"https://doi.org/10.1007/s00493-026-00224-z","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathcal {G}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>G</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be a hypergraph whose edges are colored. An <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(alpha ,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> - <jats:italic>detachment</jats:italic> of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathcal {G}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>G</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is a hypergraph obtained by splitting a vertex <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$alpha $$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> into <jats:italic>n</jats:italic> vertices, say <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$alpha _1,dots ,alpha _n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , and sharing the incident edges among the subvertices. A detachment is <jats:italic>fair</jats:italic> if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this paper we solve an open problem from the 1970s by finding necessary and sufficient conditions under which a <jats:italic>k</jats:italic> -edge-colored hypergraph <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathcal {G}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>G</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> has a fair detachment in which each color class is connected. Previously, this was not even known for the case when <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathcal {G}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>G</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions (in Part I), and completing partial regular combinatorial structures (in Part II).","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"133 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148883444","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":"Sunflowers and Ramsey Problems for Restricted Intersections","authors":"Barnabás Janzer, Zhihan Jin, Benny Sudakov, Kewen Wu","doi":"10.1007/s00493-026-00222-1","DOIUrl":"https://doi.org/10.1007/s00493-026-00222-1","url":null,"abstract":"Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 years. In this paper, we study the following Ramsey version of these problems. Given a set <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mo>⊆</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Lsubseteq {0,dots ,k-1}$$end{document}</tex-math></alternatives></inline-formula> and a family <inline-formula><alternatives><mml:math><mml:mi mathvariant=\"script\">F</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal {F}$$end{document}</tex-math></alternatives></inline-formula> of <italic>k</italic>-element sets which does not contain a sunflower with <italic>m</italic> petals whose kernel size is in <italic>L</italic>, how large a subfamily of <inline-formula><alternatives><mml:math><mml:mi mathvariant=\"script\">F</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal {F}$$end{document}</tex-math></alternatives></inline-formula> can we find in which no pair has intersection size in <italic>L</italic>? We give matching upper and lower bounds, determining the dependence on <italic>m</italic> for all <italic>k</italic> and <italic>L</italic>. This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of Füredi’s celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in Füredi’s result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148715924","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 Counterexample to a Conjecture of Lovász","authors":"Alexander Clow, Penny Haxell, Bojan Mohar","doi":"10.1007/s00493-026-00220-3","DOIUrl":"https://doi.org/10.1007/s00493-026-00220-3","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"116 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408883","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":"Ordered Ramsey Numbers of Powers of Paths","authors":"António Girão, Barnabás Janzer, Oliver Janzer","doi":"10.1007/s00493-026-00223-0","DOIUrl":"https://doi.org/10.1007/s00493-026-00223-0","url":null,"abstract":"Given two vertex-ordered graphs &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the ordered Ramsey number &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;R&lt;/mml:mi&gt;&lt;mml:mo&gt;&lt;&lt;/mml:mo&gt;&lt;/mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$R_&lt;(H_1,H_2)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is the smallest &lt;italic&gt;N&lt;/italic&gt; such that whenever the edges of a vertex-ordered complete graph &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;K&lt;/mml:mi&gt;&lt;mml:mi&gt;N&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$K_N$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; are red/blue-coloured, then there is a red (ordered) copy of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; or a blue (ordered) copy of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;H&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. Let &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msubsup&gt;&lt;mml:mi&gt;P&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;m","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"250 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148612831","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 fractional Helly number for separable convexity spaces","authors":"Andreas F. Holmsen, Zuzana Patáková","doi":"10.1007/s00493-026-00221-2","DOIUrl":"https://doi.org/10.1007/s00493-026-00221-2","url":null,"abstract":"A &lt;italic&gt;convex lattice set&lt;/italic&gt; in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;Z&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {Z}^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is the intersection of a convex set in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with the integer lattice &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;Z&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {Z}^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. A classical theorem of Doignon states that the &lt;italic&gt;Helly number&lt;/italic&gt; of &lt;italic&gt;d&lt;/italic&gt;-dimensional convex lattice sets equals &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, exponentially larger than the Helly number &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d+1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of ordinary convex sets in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. By contrast, a remarkable theorem of Bár","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"36 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148612832","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":"Volume Rigidity of Simplicial Manifolds","authors":"James Cruickshank, Bill Jackson, Shin-ichi Tanigawa","doi":"10.1007/s00493-026-00218-x","DOIUrl":"https://doi.org/10.1007/s00493-026-00218-x","url":null,"abstract":"Classical results of Cauchy [&lt;xref ref-type=\"bibr\"&gt;4&lt;/xref&gt;] and Dehn [&lt;xref ref-type=\"bibr\"&gt;5&lt;/xref&gt;] imply that the 1-skeleton of a convex simplicial polyhedron &lt;italic&gt;P&lt;/italic&gt; is rigid, i.e. every continuous motion of the vertices of &lt;italic&gt;P&lt;/italic&gt; in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mn&gt;3&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb R}^3$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; which preserves its edge lengths results in a polyhedron which is congruent to &lt;italic&gt;P&lt;/italic&gt;. This result was extended to convex simplicial polytopes in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb R}^d$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for all &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;3&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$dge 3$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; by Whiteley [&lt;xref ref-type=\"bibr\"&gt;16&lt;/xref&gt;], and to generic realisations of 1-skeletons of simplicial &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(d-1)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-manifolds in &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb R}^{d}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; by Kalai [&lt;xref ref-type=\"bibr\"&gt;8&lt;/xref&gt;] for &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;d&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;4&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;t","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"20 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288225","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":"Matroids are Equitable","authors":"Hannaneh Akrami, Siyue Liu, Roshan Raj, László A. Végh","doi":"10.1007/s00493-026-00217-y","DOIUrl":"https://doi.org/10.1007/s00493-026-00217-y","url":null,"abstract":"We show that if the ground set of a matroid can be partitioned into &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;mml:mo&gt;≥&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$kge 2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; bases, then for any given subset &lt;italic&gt;S&lt;/italic&gt; of the ground set, there is a partition into &lt;italic&gt;k&lt;/italic&gt; bases such that the sizes of the intersections of the bases with &lt;italic&gt;S&lt;/italic&gt; may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szabó (Electron. J. Comb. 2011) in the affirmative. We also investigate equitable splittings of two disjoint sets &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S_1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, and show that there is a partition into &lt;italic&gt;k&lt;/italic&gt; bases such that the sizes of the intersections with &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S_1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; may differ by at most one and the sizes of the intersections with &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;S&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$S_2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; may differ by at most two; this is the best one can hope for arbitrary matroids. We also derive applications of this result to matroid-constrained fair division problems. We show that there exists a matroid-constrained allocation that is envy-free up to one item if the valuations are identical and tri-valued additive. We also sho","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"14 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288243","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":"Tight Colorful No-Dimensional Tverberg Theorem","authors":"Polina Barabanshchikova, Grigory Ivanov, Alexander Polyanskii","doi":"10.1007/s00493-026-00219-w","DOIUrl":"https://doi.org/10.1007/s00493-026-00219-w","url":null,"abstract":"We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius &lt;italic&gt;R&lt;/italic&gt; such that, for any pairwise disjoint &lt;italic&gt;k&lt;/italic&gt;-element subsets &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mo&gt;⋯&lt;/mml:mo&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q_1,dots ,Q_n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of a normed space, there exists a partition of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;∪&lt;/mml:mo&gt;&lt;mml:mo&gt;⋯&lt;/mml:mo&gt;&lt;mml:mo&gt;∪&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Q&lt;/mml:mi&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Q_1cup cdots cup Q_n$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; into disjoint transversals &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;{&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;P&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mo&gt;⋯&lt;/mml:mo&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;P&lt;/mml:mi&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mo stretchy=\"false\"&gt;}&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${P_1,dots ,P_k}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for which a ball of radius &lt;italic&gt;R&lt;/italic&gt; intersects the convex hull of each &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;P&lt;/mml:mi&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$P_i$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo&gt;≤&lt;/mml:mo&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;mml:mo&gt;≤&lt;/mml:mo&gt;&lt;mml:mi&gt;k&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$1le ile k$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;). Our methods are determ","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355776","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":"Towards an Optimal Hypergraph Container Lemma","authors":"Marcelo Campos, Wojciech Samotij","doi":"10.1007/s00493-026-00214-1","DOIUrl":"https://doi.org/10.1007/s00493-026-00214-1","url":null,"abstract":"The hypergraph container lemma is a powerful tool in probabilistic combinatorics that has found many applications since it was first proved a decade ago. Roughly speaking, it asserts that the family of independent sets of every uniform hypergraph can be covered by a small number of almost-independent sets, called containers. In this article, we formulate and prove two new versions of the lemma that display the following three attractive features. First, they both admit short proofs that have surprising connections to other well-studied topics in probabilistic combinatorics. Second, they use alternative notions of almost-independence in order to describe the containers. Third, they yield improved dependence of the number of containers on the uniformity of the hypergraph, hitting a natural barrier for second-moment-type approaches.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"135 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355777","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 Universal Threshold for Geometric Embeddings of Trees","authors":"Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros","doi":"10.1007/s00493-026-00216-z","DOIUrl":"https://doi.org/10.1007/s00493-026-00216-z","url":null,"abstract":"A graph &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G=(V,E)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is &lt;italic&gt;geometrically embeddable&lt;/italic&gt; into a normed space &lt;italic&gt;X&lt;/italic&gt; when there is a mapping &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;ζ&lt;/mml:mi&gt;&lt;mml:mo&gt;:&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;→&lt;/mml:mo&gt;&lt;mml:mi&gt;X&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$zeta :Vrightarrow X$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; such that &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;‖&lt;/mml:mo&gt;&lt;mml:mi&gt;ζ&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;v&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mi&gt;ζ&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;w&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;‖&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;X&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mo&gt;⩽&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Vert zeta (v)-zeta (w)Vert _Xleqslant 1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; if and only if &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;{&lt;/mml:mo&gt;&lt;mml:mi&gt;v&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;w&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;}&lt;/mml:mo&gt;&lt;mml:mo&gt;∈&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${v,w}in E$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, for all distinct &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;v&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;w&lt;/mml:mi&gt;&lt;mml:mo&gt;∈&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$v,win V$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. Our ","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"242 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288244","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}