Pub Date : 2025-10-10DOI: 10.1007/s00493-025-00184-w
József Balogh, Michael C. Wigal
Let <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></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}$$r ge 3$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_184_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula> be fixed and <italic>G</italic> be an <italic>n</italic>-vertex graph. A long-standing conjecture of Győri states that if <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>e</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>k</mml:mi></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}$$e(G) = t_{r-1}(n) + k$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_184_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, where <inline-formula><alternatives><mml:math><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></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}$$t_{r-1}(n)$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_184_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula> denotes the number of edges of the Turán graph on <italic>n</italic> vertices and <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></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}$$r - 1$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_184_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula> parts, then <italic>G</italic> has at least <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy="
Let r≥3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$r ge 3$$end{document} be fixed and G be an n-vertex graph. A long-standing conjecture of Győri states that if e(G)=tr-1(n)+kdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$e(G) = t_{r-1}(n) + k$$end{document}, where tr-1(n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$t_{r-1}(n)$$end{document} denotes the number of edges of the Turán graph on n vertices and r-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$r - 1$$end{document} parts, then G has at least (2-o(1))k/rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(2 - o(1))k/r$$end{document} edge disjoint r-cliques. We prove this conjecture.
{"title":"Packing edge disjoint cliques in graphs","authors":"József Balogh, Michael C. Wigal","doi":"10.1007/s00493-025-00184-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00184-w","url":null,"abstract":"Let <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></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}$$r ge 3$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_184_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> be fixed and <italic>G</italic> be an <italic>n</italic>-vertex graph. A long-standing conjecture of Győri states that if <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>e</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>k</mml:mi></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}$$e(G) = t_{r-1}(n) + k$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_184_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>, where <inline-formula><alternatives><mml:math><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></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}$$t_{r-1}(n)$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_184_Article_IEq3.gif\"></inline-graphic></alternatives></inline-formula> denotes the number of edges of the Turán graph on <italic>n</italic> vertices and <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></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}$$r - 1$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_184_Article_IEq4.gif\"></inline-graphic></alternatives></inline-formula> parts, then <italic>G</italic> has at least <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914891","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}
Pub Date : 2025-10-10DOI: 10.1007/s00493-025-00182-y
Jindřich Zapletal
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles in Euclidean plane has countable chromatic number, while there is no Vitali set.
{"title":"Triangles and Vitali Sets","authors":"Jindřich Zapletal","doi":"10.1007/s00493-025-00182-y","DOIUrl":"https://doi.org/10.1007/s00493-025-00182-y","url":null,"abstract":"It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles in Euclidean plane has countable chromatic number, while there is no Vitali set.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"77 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914892","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}
Pub Date : 2025-10-10DOI: 10.1007/s00493-025-00181-z
Rutger Campbell, Jim Geelen, Matthew E. Kroeker
Melchior’s inequality implies that the average line-length in a simple, rank-3, real-representable matroid is less than 3. A similar result holds for complex-representable matroids, using Hirzebruch’s inequality, but with a weaker bound of 4. We show that the average plane-size in a simple, rank-4, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer k, in complex-representable matroids with rank at least 2k-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2k-1$$end{document}, the average size of a rank-k flat is bounded above by a constant depending only on k. Finally, we prove that, for any integer r⩾2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$rgeqslant 2$$end{document}, the average flat-size in rank-r complex-representable matroids is bounded above by a constant depending only on r. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-k flats in a complex-representable matroid.
Melchior’s inequality implies that the average line-length in a simple, rank-3, real-representable matroid is less than 3. A similar result holds for complex-representable matroids, using Hirzebruch’s inequality, but with a weaker bound of 4. We show that the average plane-size in a simple, rank-4, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer k, in complex-representable matroids with rank at least 2k-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2k-1$$end{document}, the average size of a rank-k flat is bounded above by a constant depending only on k. Finally, we prove that, for any integer r⩾2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$rgeqslant 2$$end{document}, the average flat-size in rank-r complex-representable matroids is bounded above by a constant depending only on r. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-k flats in a complex-representable matroid.
{"title":"Average plane-size in complex-representable matroids","authors":"Rutger Campbell, Jim Geelen, Matthew E. Kroeker","doi":"10.1007/s00493-025-00181-z","DOIUrl":"https://doi.org/10.1007/s00493-025-00181-z","url":null,"abstract":"Melchior’s inequality implies that the average line-length in a simple, rank-3, real-representable matroid is less than 3. A similar result holds for complex-representable matroids, using Hirzebruch’s inequality, but with a weaker bound of 4. We show that the average plane-size in a simple, rank-4, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer <italic>k</italic>, in complex-representable matroids with rank at least <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></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}$$2k-1$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_181_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula>, the average size of a rank-<italic>k</italic> flat is bounded above by a constant depending only on <italic>k</italic>. Finally, we prove that, for any integer <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>r</mml:mi><mml:mo>⩾</mml:mo><mml:mn>2</mml:mn></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}$$rgeqslant 2$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_181_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>, the average flat-size in rank-<italic>r</italic> complex-representable matroids is bounded above by a constant depending only on <italic>r</italic>. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-<italic>k</italic> flats in a complex-representable matroid.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"42 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914894","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}
Pub Date : 2025-10-10DOI: 10.1007/s00493-025-00176-w
Đorđe Mitrović, Jeroen Schillewaert, Hendrik Van Maldeghem
We provide local recognition results for all infinite families of affine rank 3 graphs which are not locally a disjoint union of cliques and which are not 1-dimensional (that is, for the graphs corresponding to Liebeck’s classes (A3) up to (A10) of affine rank 3 groups). We show that the local graphs alone do not characterise the global graphs by providing counterexamples. Our principal result is that the global graph is determined by the local if one adds the condition that the number of vertices adjacent to two given vertices at distance 2 from each other is a constant. Alternative conditions for specific classes are also given.
{"title":"Local recognition of affine rank 3 graphs","authors":"Đorđe Mitrović, Jeroen Schillewaert, Hendrik Van Maldeghem","doi":"10.1007/s00493-025-00176-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00176-w","url":null,"abstract":"We provide local recognition results for all infinite families of affine rank 3 graphs which are not locally a disjoint union of cliques and which are not 1-dimensional (that is, for the graphs corresponding to Liebeck’s classes (A3) up to (A10) of affine rank 3 groups). We show that the local graphs alone do not characterise the global graphs by providing counterexamples. Our principal result is that the global graph is determined by the local if one adds the condition that the number of vertices adjacent to two given vertices at distance 2 from each other is a constant. Alternative conditions for specific classes are also given.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914893","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00173-z
Vincent Pfenninger
A <italic>k</italic>-<italic>uniform tight cycle</italic> is a <italic>k</italic>-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of <italic>k</italic> consecutive vertices in that ordering. We show that, for each <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_173_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>, the Ramsey number of the <italic>k</italic>-uniform tight cycle on <italic>kn</italic> vertices is <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(1+o(1))(k+1)n$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_173_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula>. This is an extension to all uniformities of previous results for <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 3$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_173_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula> by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 4$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_173_Article_IEq6.gif"></inline-graphic></alternatives></inline-formula> by Lo and the author and confirms a special case of a conjecture by the former set of authors. Lehel’s conjecture, which was proved by Bessy and Thomassé, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for <italic>k</italic>-uniform tight cycles. We show that, for every <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-us
A k-uniform tight cycle is a k-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of k consecutive vertices in that ordering. We show that, for each documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}, the Ramsey number of the k-uniform tight cycle on kn vertices is documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(1+o(1))(k+1)n$$end{document}. This is an extension to all uniformities of previous results for documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 3$$end{document} by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 4$$end{document} by Lo and the author and confirms a special case of a conjecture by the former set of authors. Lehel’s conjecture, which was proved by Bessy and Thomassé, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for k-uniform tight cycles. We show that, for every documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}, every red-blue edge-coloured complete k-graph on n vertices contains a red tight cycle and a blue tight cycle that are vertex-disjoint and together cover documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n - o(n)$$end{document} vertices.
{"title":"On k-uniform Tight Cycles: the Ramsey Number for documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C_{kn}^{(k)}$$end{document} and an Approximate Lehel’s Conjecture","authors":"Vincent Pfenninger","doi":"10.1007/s00493-025-00173-z","DOIUrl":"https://doi.org/10.1007/s00493-025-00173-z","url":null,"abstract":"A <italic>k</italic>-<italic>uniform tight cycle</italic> is a <italic>k</italic>-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of <italic>k</italic> consecutive vertices in that ordering. We show that, for each <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_173_Article_IEq3.gif\"></inline-graphic></alternatives></inline-formula>, the Ramsey number of the <italic>k</italic>-uniform tight cycle on <italic>kn</italic> vertices is <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(1+o(1))(k+1)n$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_173_Article_IEq4.gif\"></inline-graphic></alternatives></inline-formula>. This is an extension to all uniformities of previous results for <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 3$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_173_Article_IEq5.gif\"></inline-graphic></alternatives></inline-formula> by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k = 4$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_173_Article_IEq6.gif\"></inline-graphic></alternatives></inline-formula> by Lo and the author and confirms a special case of a conjecture by the former set of authors. Lehel’s conjecture, which was proved by Bessy and Thomassé, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for <italic>k</italic>-uniform tight cycles. We show that, for every <inline-formula><alternatives><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k ge 3$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-us","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"20 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914901","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00174-y
Nikolai Terekhov, Maksim Zhukovskii
Given a graph <italic>F</italic> and a positive integer <italic>n</italic>, the weak <italic>F</italic>-saturation number <inline-formula><alternatives><mml:math><mml:mrow><mml:mtext>wsat</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>F</mml:mi><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}$${textrm{wsat}}(K_n,F)$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_174_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula> is the minimum number of edges in a graph <italic>H</italic> on <italic>n</italic> vertices such that the edges missing in <italic>H</italic> can be added, one at a time, so that every edge creates a copy of <italic>F</italic>. Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. Let <italic>W</italic> be a vector space spanned by vectors <italic>w</italic>(<italic>e</italic>) assigned to edges <italic>e</italic> of <inline-formula><alternatives><mml:math><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></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}$$K_n$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_174_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>. Suppose that, for every copy <inline-formula><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>F</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>⊂</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></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}$$F'subset K_n$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_174_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula> of <italic>F</italic>, there exist non-zero scalars <inline-formula><alternatives><mml:math><mml:msub><mml:mi>λ</mml:mi><mml:mi>e</mml:mi></mml:msub></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}$$lambda _e$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_174_Article_IEq4.gi
Given a graph F and a positive integer n, the weak F-saturation number wsat(Kn,F)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${textrm{wsat}}(K_n,F)$$end{document} is the minimum number of edges in a graph H on n vertices such that the edges missing in H can be added, one at a time, so that every edge creates a copy of F. Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. Let W be a vector space spanned by vectors w(e) assigned to edges e of Kndocumentclass[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}. Suppose that, for every copy F′⊂Kndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$F'subset K_n$$end{document} of F, there exist non-zero scalars λedocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda _e$$end{document}, e∈E(F′)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ein E(F')$$end{document}, satisfying ∑e∈E(F′)λew(e)=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$sum _{ein E(F')}lambda _e w(e)=0$$end{document}. Then dimW≤wsat(Kn,F)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$textrm{dim}Wle {textrm{wsat}}(K_n,F)$$end{document}. In this paper, we prove limitations of this approach: we find infinitely many F such that, for every vector space W as above, dimW<wsat(Kn,F)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$textrm{dim}W<{textrm{wsat}}(K_n,F)$$end{document}. We also introduce a modification of this approach that yields tight lower bounds even when the original direct approach is insufficient. Finally, we generalise our results to random graphs, complete multipartite graphs, and h
{"title":"Weak saturation rank: a failure of the linear algebraic approach to weak saturation","authors":"Nikolai Terekhov, Maksim Zhukovskii","doi":"10.1007/s00493-025-00174-y","DOIUrl":"https://doi.org/10.1007/s00493-025-00174-y","url":null,"abstract":"Given a graph <italic>F</italic> and a positive integer <italic>n</italic>, the weak <italic>F</italic>-saturation number <inline-formula><alternatives><mml:math><mml:mrow><mml:mtext>wsat</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>F</mml:mi><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}$${textrm{wsat}}(K_n,F)$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_174_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> is the minimum number of edges in a graph <italic>H</italic> on <italic>n</italic> vertices such that the edges missing in <italic>H</italic> can be added, one at a time, so that every edge creates a copy of <italic>F</italic>. Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. Let <italic>W</italic> be a vector space spanned by vectors <italic>w</italic>(<italic>e</italic>) assigned to edges <italic>e</italic> of <inline-formula><alternatives><mml:math><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></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}$$K_n$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_174_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>. Suppose that, for every copy <inline-formula><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>F</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>⊂</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></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}$$F'subset K_n$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_174_Article_IEq3.gif\"></inline-graphic></alternatives></inline-formula> of <italic>F</italic>, there exist non-zero scalars <inline-formula><alternatives><mml:math><mml:msub><mml:mi>λ</mml:mi><mml:mi>e</mml:mi></mml:msub></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}$$lambda _e$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_174_Article_IEq4.gi","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"43 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914900","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00178-8
Ron Aharoni, Eli Berger, He Guo, Dani Kotlar
It is known that in matroids the difference between the chromatic number and the fractional chromatic number is smaller than 1, and that the list chromatic number is equal to the chromatic number. We investigate the gap within these pairs of parameters for hypergraphs that are the intersection of a given number k of matroids. We prove that in such hypergraphs the list chromatic number is at most k times the chromatic number and at most 2k-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2k-1$$end{document} times the maximum chromatic number among the k matroids. We study the relationship between three polytopes associated with k-sets of matroids, and connect them to bounds on the fractional chromatic number of the intersection of the members of the k-set. This also connects to bounds on the matroidal matching and covering number of the intersection of the members of the k-set. The tools used are in part topological.
It is known that in matroids the difference between the chromatic number and the fractional chromatic number is smaller than 1, and that the list chromatic number is equal to the chromatic number. We investigate the gap within these pairs of parameters for hypergraphs that are the intersection of a given number k of matroids. We prove that in such hypergraphs the list chromatic number is at most k times the chromatic number and at most 2k-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2k-1$$end{document} times the maximum chromatic number among the k matroids. We study the relationship between three polytopes associated with k-sets of matroids, and connect them to bounds on the fractional chromatic number of the intersection of the members of the k-set. This also connects to bounds on the matroidal matching and covering number of the intersection of the members of the k-set. The tools used are in part topological.
{"title":"Coloring, list coloring, and fractional coloring in intersections of matroids","authors":"Ron Aharoni, Eli Berger, He Guo, Dani Kotlar","doi":"10.1007/s00493-025-00178-8","DOIUrl":"https://doi.org/10.1007/s00493-025-00178-8","url":null,"abstract":"It is known that in matroids the difference between the chromatic number and the fractional chromatic number is smaller than 1, and that the list chromatic number is equal to the chromatic number. We investigate the gap within these pairs of parameters for hypergraphs that are the intersection of a given number <italic>k</italic> of matroids. We prove that in such hypergraphs the list chromatic number is at most <italic>k</italic> times the chromatic number and at most <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></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}$$2k-1$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_178_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> times the maximum chromatic number among the <italic>k</italic> matroids. We study the relationship between three polytopes associated with <italic>k</italic>-sets of matroids, and connect them to bounds on the fractional chromatic number of the intersection of the members of the <italic>k</italic>-set. This also connects to bounds on the matroidal matching and covering number of the intersection of the members of the <italic>k</italic>-set. The tools used are in part topological.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914899","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00179-7
Jan Hubička, Colin Jahel, Matěj Konečný, Marcin Sabok
We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite n-partite tournaments for n∈{2,3,…,ω}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n in {2,3,ldots ,omega }$$end{document}, and for the class of finite semigeneric tournaments. We also prove that the generic ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$omega $$end{document}-partite tournament and the generic semigeneric tournament have ample generics.
We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite n-partite tournaments for n∈{2,3,…,ω}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n in {2,3,ldots ,omega }$$end{document}, and for the class of finite semigeneric tournaments. We also prove that the generic ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$omega $$end{document}-partite tournament and the generic semigeneric tournament have ample generics.
{"title":"Extension Property for Partial Automorphisms of the n-partite and Semigeneric Tournaments","authors":"Jan Hubička, Colin Jahel, Matěj Konečný, Marcin Sabok","doi":"10.1007/s00493-025-00179-7","DOIUrl":"https://doi.org/10.1007/s00493-025-00179-7","url":null,"abstract":"We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite <italic>n</italic>-partite tournaments for <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>ω</mml:mi><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}$$n in {2,3,ldots ,omega }$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_179_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula>, and for the class of finite semigeneric tournaments. We also prove that the generic <inline-formula><alternatives><mml:math><mml:mi>ω</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}$$omega $$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_179_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>-partite tournament and the generic semigeneric tournament have ample generics.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"84 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914902","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00180-0
Lars Becker, Paata Ivanisvili, Dmitry Krachun, José Madrid
We show that for all <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mi>d</mml:mi></mml:msup></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}$$A, B subseteq {0,1,2}^{d}$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_180_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula> we have <disp-formula><alternatives><mml:math display="block"><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>⩾</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>log</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mfrac></mml:msup><mml:mo>.</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}$$|A+B|geqslant (|A||B|)^frac{log 5}{2log 3}.$$end{document}</tex-math><graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_180_Article_Equ37.gif"></graphic></alternatives></disp-formula>We also show that for all finite <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></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}$$A,B subset mathbb {Z}^{d}$$end{document}</tex-math><inline-graphic mime-subtype="GIF" specific-use="web" xlink:href="493_2025_180_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, and any <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>V</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12
We show that for all A,B⊆{0,1,2}ddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A, B subseteq {0,1,2}^{d}$$end{document} we have |A+B|⩾(|A||B|)log52log3.documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$|A+B|geqslant (|A||B|)^frac{log 5}{2log 3}.$$end{document}We also show that for all finite A,B⊂Zddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A,B subset mathbb {Z}^{d}$$end{document}, and any V⊆{0,1}ddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$V subseteq {0,1}^{d}$$end{document} the inequality |A+B+V|⩾|A|1/p|B|1/q|V|log2(p1/pq1/q)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$|A+B+V|geqslant |A|^{1/p}|B|^{1/q}|V|^{log _{2}(p^{1/p}q^{1/q})}$$end{document}holds for all p∈(1,∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p in (1, infty )$$end{document}, where q=pp-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$q=frac{p}{p-1}$$end{document} is the conjugate exponent of p. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems.
{"title":"Discrete Brunn–Minkowski inequality for subsets of the cube","authors":"Lars Becker, Paata Ivanisvili, Dmitry Krachun, José Madrid","doi":"10.1007/s00493-025-00180-0","DOIUrl":"https://doi.org/10.1007/s00493-025-00180-0","url":null,"abstract":"We show that for all <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mi>d</mml:mi></mml:msup></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}$$A, B subseteq {0,1,2}^{d}$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_180_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> we have <disp-formula><alternatives><mml:math display=\"block\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>⩾</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>log</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mfrac></mml:msup><mml:mo>.</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}$$|A+B|geqslant (|A||B|)^frac{log 5}{2log 3}.$$end{document}</tex-math><graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_180_Article_Equ37.gif\"></graphic></alternatives></disp-formula>We also show that for all finite <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></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}$$A,B subset mathbb {Z}^{d}$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_180_Article_IEq2.gif\"></inline-graphic></alternatives></inline-formula>, and any <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>V</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914897","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}
Pub Date : 2025-09-09DOI: 10.1007/s00493-025-00177-9
Tung Nguyen, Alex Scott, Paul Seymour
When H is a forest, the Gyárfás-Sumner conjecture implies that every graph G with no induced subgraph isomorphic to H and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph G has a stable set of size |G|1-o(1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$|G|^{1-o(1)}$$end{document}. If H is not a forest, there need not be such a stable set. Second, we prove that when H is a “multibroom”, there is a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a “fractional colouring” version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.
When H is a forest, the Gyárfás-Sumner conjecture implies that every graph G with no induced subgraph isomorphic to H and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph G has a stable set of size |G|1-o(1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$|G|^{1-o(1)}$$end{document}. If H is not a forest, there need not be such a stable set. Second, we prove that when H is a “multibroom”, there is a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a “fractional colouring” version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.
{"title":"Trees and near-linear stable sets","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1007/s00493-025-00177-9","DOIUrl":"https://doi.org/10.1007/s00493-025-00177-9","url":null,"abstract":"When <italic>H</italic> is a forest, the Gyárfás-Sumner conjecture implies that every graph <italic>G</italic> with no induced subgraph isomorphic to <italic>H</italic> and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph <italic>G</italic> has a stable set of size <inline-formula><alternatives><mml:math><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></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}$$|G|^{1-o(1)}$$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"493_2025_177_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula>. If <italic>H</italic> is not a forest, there need not be such a stable set. Second, we prove that when <italic>H</italic> is a “multibroom”, there is a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a “fractional colouring” version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"40 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914898","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：