{"title":"随机多项式图的随机Turán问题","authors":"Sam Spiro","doi":"10.1002/jgt.23015","DOIUrl":null,"url":null,"abstract":"<p>Bukh and Conlon used random polynomial graphs to give effective lower bounds on <math>\n <semantics>\n <mrow>\n <mtext>ex</mtext>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n <mo>,</mo>\n <msup>\n <mi>T</mi>\n <mi>ℓ</mi>\n </msup>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{ex}(n,{{\\mathscr{T}}}^{\\ell })$</annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n <msup>\n <mi>T</mi>\n <mi>ℓ</mi>\n </msup>\n </mrow>\n <annotation> ${{\\mathscr{T}}}^{\\ell }$</annotation>\n </semantics></math> is the <math>\n <semantics>\n <mrow>\n <mi>ℓ</mi>\n </mrow>\n <annotation> $\\ell $</annotation>\n </semantics></math>th power of a balanced rooted tree <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math>. We extend their result to give effective lower bounds on <math>\n <semantics>\n <mrow>\n <mtext>ex</mtext>\n <mrow>\n <mo>(</mo>\n <mrow>\n <msub>\n <mi>G</mi>\n <mrow>\n <mi>n</mi>\n <mo>,</mo>\n <mi>p</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <msup>\n <mi>T</mi>\n <mi>ℓ</mi>\n </msup>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{ex}({G}_{n,p},{{\\mathscr{T}}}^{\\ell })$</annotation>\n </semantics></math>, which is the maximum number of edges in a <math>\n <semantics>\n <mrow>\n <msup>\n <mi>T</mi>\n <mi>ℓ</mi>\n </msup>\n </mrow>\n <annotation> ${{\\mathscr{T}}}^{\\ell }$</annotation>\n </semantics></math>-free subgraph of the random graph <math>\n <semantics>\n <mrow>\n <msub>\n <mi>G</mi>\n <mrow>\n <mi>n</mi>\n <mo>,</mo>\n <mi>p</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${G}_{n,p}$</annotation>\n </semantics></math>. Analogous bounds for generalized Turán numbers in random graphs are also proven.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"105 2","pages":"192-208"},"PeriodicalIF":0.9000,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Random polynomial graphs for random Turán problems\",\"authors\":\"Sam Spiro\",\"doi\":\"10.1002/jgt.23015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Bukh and Conlon used random polynomial graphs to give effective lower bounds on <math>\\n <semantics>\\n <mrow>\\n <mtext>ex</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n <mo>,</mo>\\n <msup>\\n <mi>T</mi>\\n <mi>ℓ</mi>\\n </msup>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{ex}(n,{{\\\\mathscr{T}}}^{\\\\ell })$</annotation>\\n </semantics></math>, where <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>T</mi>\\n <mi>ℓ</mi>\\n </msup>\\n </mrow>\\n <annotation> ${{\\\\mathscr{T}}}^{\\\\ell }$</annotation>\\n </semantics></math> is the <math>\\n <semantics>\\n <mrow>\\n <mi>ℓ</mi>\\n </mrow>\\n <annotation> $\\\\ell $</annotation>\\n </semantics></math>th power of a balanced rooted tree <math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math>. We extend their result to give effective lower bounds on <math>\\n <semantics>\\n <mrow>\\n <mtext>ex</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <msub>\\n <mi>G</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>,</mo>\\n <mi>p</mi>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <msup>\\n <mi>T</mi>\\n <mi>ℓ</mi>\\n </msup>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{ex}({G}_{n,p},{{\\\\mathscr{T}}}^{\\\\ell })$</annotation>\\n </semantics></math>, which is the maximum number of edges in a <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>T</mi>\\n <mi>ℓ</mi>\\n </msup>\\n </mrow>\\n <annotation> ${{\\\\mathscr{T}}}^{\\\\ell }$</annotation>\\n </semantics></math>-free subgraph of the random graph <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>G</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>,</mo>\\n <mi>p</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${G}_{n,p}$</annotation>\\n </semantics></math>. Analogous bounds for generalized Turán numbers in random graphs are also proven.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"105 2\",\"pages\":\"192-208\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-08-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23015\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23015","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Random polynomial graphs for random Turán problems
Bukh and Conlon used random polynomial graphs to give effective lower bounds on , where is the th power of a balanced rooted tree . We extend their result to give effective lower bounds on , which is the maximum number of edges in a -free subgraph of the random graph . Analogous bounds for generalized Turán numbers in random graphs are also proven.
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 .
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