On the (6,4)-problem of Brown, Erdős, and Sós

Proceedings of the American Mathematical Society, Series B Pub Date : 2024-06-04 DOI:10.1090/bproc/170

Stefan Glock, Felix Joos, Jaehoon Kim, Marcus Kühn, Lyuben Lichev, Oleg Pikhurko

{"title":"On the (6,4)-problem of Brown, Erdős, and Sós","authors":"Stefan Glock, Felix Joos, Jaehoon Kim, Marcus Kühn, Lyuben Lichev, Oleg Pikhurko","doi":"10.1090/bproc/170","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript left-parenthesis r right-parenthesis Baseline left-parenthesis n semicolon s comma k right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>;</mml:mo>\n <mml:mi>s</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f^{(r)}(n;s,k)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be the maximum number of edges of an <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\">\n <mml:semantics>\n <mml:mi>r</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">r</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-uniform hypergraph on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> vertices not containing a subgraph with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> edges and at most <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s\">\n <mml:semantics>\n <mml:mi>s</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">s</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> vertices. In 1973, Brown, Erdős, and Sós conjectured that the limit <disp-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit Underscript n right-arrow normal infinity Endscripts n Superscript negative 2 f Superscript left-parenthesis 3 right-parenthesis Baseline left-parenthesis n semicolon k plus 2 comma k right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>n</mml:mi>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:msup>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>;</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\begin{equation*} \\lim _{n\\to \\infty } n^{-2} f^{(3)}(n;k+2,k) \\end{equation*}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</disp-formula>\n exists for all <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and confirmed it for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">k=2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Recently, Glock showed this for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">k=3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We settle the next open case, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals 4\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">k=4</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, by showing that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript left-parenthesis 3 right-parenthesis Baseline left-parenthesis n semicolon 6 comma 4 right-parenthesis equals left-parenthesis seven thirty-sixths plus o left-parenthesis 1 right-parenthesis right-parenthesis n squared\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>;</mml:mo>\n <mml:mn>6</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mfrac>\n <mml:mn>7</mml:mn>\n <mml:mn>36</mml:mn>\n </mml:mfrac>\n <mml:mo>+</mml:mo>\n <mml:mi>o</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msup>\n <mml:mi>n</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f^{(3)}(n;6,4)=\\left (\\frac {7}{36}+o(1)\\right )n^2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n right-arrow normal infinity\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n\\to \\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. More generally, for all <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k element-of StartSet 3 comma 4 EndSet\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">k\\in \\{3,4\\}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r greater-than-or-equal-to 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>r</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">r\\ge 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t element-of left-bracket 2 comma r minus 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>t</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">t\\in [2,r-1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we compute the value of the limit <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit Underscript n right-arrow normal infinity Endscripts n Superscript negative t f Superscript left-parenthesis r right-parenthesis Baseline left-parenthesis n semicolon k left-parenthesis r minus t right-parenthesis plus t comma k right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>n</mml:mi>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:msup>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−</mml:mo>\n <mml:mi>t</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>r</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>;</mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mm","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"87 12","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/170","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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Abstract

Let f ( r ) ( n ; s , k ) f^{(r)}(n;s,k) be the maximum number of edges of an r r -uniform hypergraph on n n vertices not containing a subgraph with k k edges and at most s s vertices. In 1973, Brown, Erdős, and Sós conjectured that the limit lim n → ∞ n − 2 f ( 3 ) ( n ; k + 2 , k ) \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all k k and confirmed it for k = 2 k=2 . Recently, Glock showed this for k = 3 k=3 . We settle the next open case, k = 4 k=4 , by showing that f ( 3 ) ( n ; 6 , 4 ) = ( 7 36 + o ( 1 ) ) n 2 f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right )n^2 as n → ∞ n\to \infty . More generally, for all k ∈ { 3 , 4 } k\in \{3,4\} , r ≥ 3 r\ge 3 and t ∈ [ 2 , r − 1 ] t\in [2,r-1] , we compute the value of the limit lim n → ∞ n − t f ( r ) ( n ; k (

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关于布朗、厄尔多斯和索斯的 (6,4)- 问题

设 f ( r ) ( n ; s , k ) f^{(r)}(n;s,k) 是一个 n n 个顶点上的 r r 个均匀超图的最大边数，该超图不包含一个有 k k 条边、最多有 s s 个顶点的子图。1973 年，布朗、厄尔多斯和索斯猜想，极限 lim n → ∞ n - 2 f ( 3 ) ( n ; k + 2 , k ) （开始{公式*}）。\lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} 对于所有 k k 都存在，并且在 k = 2 k=2 时得到了证实。最近，格洛克证明了 k = 3 k=3 的情况。我们通过证明 f ( 3 ) ( n ; 6 , 4 ) = ( 7 36 + o ( 1 ) ) n 2 f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right )n^2 as n → ∞ n\to \infty 来解决下一个未知情况，即 k = 4 k=4 。更一般地说，对于所有 k∈ { 3 , 4 } k\in \{3,4\} ，r ≥ 3\rge ≥ 3 \rge ≥ 3 \rge ≥ 3 \rge 。 , r ≥ 3 r\ge 3 and t ∈ [ 2 , r - 1 ] t\in [2,r-1] , 我们计算极限值 lim n → ∞ n - t f ( r ) ( n ; k (

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