{"title":"On non-degenerate Turán problems for expansions","authors":"Dániel Gerbner","doi":"10.1016/j.ejc.2024.104071","DOIUrl":null,"url":null,"abstract":"<div><div>The <span><math><mi>r</mi></math></span>-uniform expansion <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span> of a graph <span><math><mi>F</mi></math></span> is obtained by enlarging each edge with <span><math><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></math></span> new vertices such that altogether we use <span><math><mrow><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> new vertices. Two simple lower bounds on the largest number <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>r</mi></math></span>-edges in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span>-free <span><math><mi>r</mi></math></span>-graphs are <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> (in the case <span><math><mi>F</mi></math></span> is not a star) and <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which is the largest number of <span><math><mi>r</mi></math></span>-cliques in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs. We prove that <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The proof comes with a structure theorem that we use to determine <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> exactly for some graphs <span><math><mi>F</mi></math></span>, every <span><math><mrow><mi>r</mi><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001562/pdfft?md5=86fa8d5991cc3c3ff302bc8fdbd50279&pid=1-s2.0-S0195669824001562-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001562","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The -uniform expansion of a graph is obtained by enlarging each edge with new vertices such that altogether we use new vertices. Two simple lower bounds on the largest number of -edges in -free -graphs are (in the case is not a star) and , which is the largest number of -cliques in -vertex -free graphs. We prove that . The proof comes with a structure theorem that we use to determine exactly for some graphs , every and sufficiently large .
图 F 的 r-uniform 扩展 F(r)+ 是通过用 r-2 个新顶点扩大每条边而得到的,这样我们总共使用了 (r-2)|E(F)| 个新顶点。关于无 F(r)+ r 图中 r 边的最大数量 exr(n,F(r)+) 的两个简单下限是 Ω(nr-1)(在 F 不是星形的情况下)和 ex(n,Kr,F),后者是无 n 个顶点的 F 图中 r 簇的最大数量。我们证明,exr(n,F(r)+)=ex(n,Kr,F)+O(nr-1)。该证明包含一个结构定理,我们用它来精确确定某些图 F、每个 r<χ(F)和足够大的 n 的 exr(n,F(r)+)。
期刊介绍:
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.