Turán数字T(n,5,3)$ T(n,5,3)$和没有诱导5 -环的图

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2023-08-07 DOI:10.1002/jgt.23021

Iliya Bluskov, Jan de Heer, Alexander Sidorenko

{"title":"Turán数字T(n,5,3)$ T(n,5,3)$和没有诱导5 -环的图","authors":"Iliya Bluskov, Jan de Heer, Alexander Sidorenko","doi":"10.1002/jgt.23021","DOIUrl":null,"url":null,"abstract":"<p>The Turán number <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $T(n,5,3)$</annotation>\n </semantics></math> is the minimum size of a system of triples out of a base set <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation> $X$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> elements such that every quintuple in <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation> $X$</annotation>\n </semantics></math> contains a triple from the system. The exact values of <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $T(n,5,3)$</annotation>\n </semantics></math> are known for <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>≤</mo>\n \n <mn>17</mn>\n </mrow>\n <annotation> $n\\le 17$</annotation>\n </semantics></math>. Turán conjectured that <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mn>2</mn>\n \n <mfenced>\n <mfrac>\n <mi>m</mi>\n \n <mn>3</mn>\n </mfrac>\n </mfenced>\n </mrow>\n <annotation> $T(2m,5,3)=2\\left(\\genfrac{}{}{0.0pt}{}{m}{3}\\right)$</annotation>\n </semantics></math>, and no counterexamples have been found so far. If this conjecture is true, then <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n <mrow>\n <mo>⌈</mo>\n <mrow>\n <mi>m</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>m</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∕</mo>\n \n <mn>6</mn>\n </mrow>\n \n <mo>⌉</mo>\n </mrow>\n </mrow>\n <annotation> $T(2m+1,5,3)\\ge \\lceil m(m-2)(2m+1)\\unicode{x02215}6\\rceil $</annotation>\n </semantics></math>. We prove the matching upper bound for all <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>=</mo>\n \n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>></mo>\n \n <mn>17</mn>\n </mrow>\n <annotation> $n=2m+1\\gt 17$</annotation>\n </semantics></math> except <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>=</mo>\n \n <mn>27</mn>\n </mrow>\n <annotation> $n=27$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"104 3","pages":"451-460"},"PeriodicalIF":0.9000,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Turán numbers \\n \\n \\n T\\n \\n (\\n \\n n\\n ,\\n 5\\n ,\\n 3\\n \\n )\\n \\n \\n $T(n,5,3)$\\n and graphs without induced 5-cycles\",\"authors\":\"Iliya Bluskov, Jan de Heer, Alexander Sidorenko\",\"doi\":\"10.1002/jgt.23021\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Turán number <math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mn>5</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $T(n,5,3)$</annotation>\\n </semantics></math> is the minimum size of a system of triples out of a base set <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n </mrow>\\n <annotation> $X$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> elements such that every quintuple in <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n </mrow>\\n <annotation> $X$</annotation>\\n </semantics></math> contains a triple from the system. The exact values of <math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mn>5</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $T(n,5,3)$</annotation>\\n </semantics></math> are known for <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>≤</mo>\\n \\n <mn>17</mn>\\n </mrow>\\n <annotation> $n\\\\le 17$</annotation>\\n </semantics></math>. Turán conjectured that <math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>m</mi>\\n \\n <mo>,</mo>\\n \\n <mn>5</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>=</mo>\\n \\n <mn>2</mn>\\n \\n <mfenced>\\n <mfrac>\\n <mi>m</mi>\\n \\n <mn>3</mn>\\n </mfrac>\\n </mfenced>\\n </mrow>\\n <annotation> $T(2m,5,3)=2\\\\left(\\\\genfrac{}{}{0.0pt}{}{m}{3}\\\\right)$</annotation>\\n </semantics></math>, and no counterexamples have been found so far. If this conjecture is true, then <math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>m</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <mn>5</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n <mrow>\\n <mo>⌈</mo>\\n <mrow>\\n <mi>m</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>m</mi>\\n \\n <mo>−</mo>\\n \\n <mn>2</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>m</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∕</mo>\\n \\n <mn>6</mn>\\n </mrow>\\n \\n <mo>⌉</mo>\\n </mrow>\\n </mrow>\\n <annotation> $T(2m+1,5,3)\\\\ge \\\\lceil m(m-2)(2m+1)\\\\unicode{x02215}6\\\\rceil $</annotation>\\n </semantics></math>. 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引用次数: 0

摘要

Turán数T（n，5，3）$T（n，5,3）$是n$n$元素的基集X$X$中的三元组系统的最小大小，使得X$X$中的每个五元组都包含来自该系统的三元组。当n≤17$n\le 17$时，T（n，5，3）$T（n、5、3）$的精确值是已知的。Turán推测T（2 m，5，3）=2m 3$T（2，5，3$T）=2\left（\genfrac{}{}{0.0pt}{}}{m}{3}\right）$，到目前为止还没有发现反例。如果这个猜想成立，则T（2m+1，5，3）≥？m（m-2）（2m+1）/6？$T（2m+1,5,3）\ge\lceil m（m-1）\unicode{x02215}6\rceil$。我们证明了除n=27$n=27$以外的所有n=2m+1>17$n=2m+1\gt 17$的匹配上界。

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Turán numbers T ( n , 5 , 3 ) $T(n,5,3)$ and graphs without induced 5-cycles

The Turán number $T (n, 5, 3)$ is the minimum size of a system of triples out of a base set $X$ of $n$ elements such that every quintuple in $X$ contains a triple from the system. The exact values of $T (n, 5, 3)$ are known for $n \leq 17$ . Turán conjectured that $T (2 m, 5, 3) = 2 (\frac{m}{3})$ , and no counterexamples have been found so far. If this conjecture is true, then $T (2 m + 1, 5, 3) \geq ⌈ m (m - 2) (2 m + 1) ∕ 6 ⌉$ . We prove the matching upper bound for all $n = 2 m + 1 > 17$ except $n = 27$ .

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .