{"title":"Star-critical Ramsey numbers involving large books","authors":"Xun Chen , Qizhong Lin , Lin Niu","doi":"10.1016/j.disc.2024.114270","DOIUrl":null,"url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi><mo>,</mo><mi>G</mi></math></span> and <em>H</em>, let <span><math><mi>F</mi><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> signify that any red/blue edge coloring of <em>F</em> contains either a red <em>G</em> or a blue <em>H</em>. The Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is defined to be the smallest integer <em>r</em> such that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span> be the book graph which consists of <em>n</em> copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> all sharing a common <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, and let <span><math><mi>G</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> be the complete <span><math><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>, <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>.</div><div>In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and sufficiently large <em>n</em>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>2</mn></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo></math></span>. This implies that the star-critical Ramsey number <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo><mo>=</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>. As a corollary, <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo><mo>=</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>k</mi></math></span> for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. This solves a problem proposed by Hao and Lin (2018) <span><span>[11]</span></span> in a stronger form.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114270"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004011","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
For graphs and H, let signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number is defined to be the smallest integer r such that . Let be the book graph which consists of n copies of all sharing a common , and let be the complete -partite graph with , and .
In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed , and sufficiently large n, . This implies that the star-critical Ramsey number . As a corollary, for and . This solves a problem proposed by Hao and Lin (2018) [11] in a stronger form.
期刊介绍:
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