{"title":"涉及大型图书的星形临界拉姆齐数字","authors":"","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":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Star-critical Ramsey numbers involving large books\",\"authors\":\"\",\"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\":null,\"pages\":null},\"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}","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}
Star-critical Ramsey numbers involving large books
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:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
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