涉及大型图书的星形临界拉姆齐数字

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-11 DOI:10.1016/j.disc.2024.114270
{"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}
引用次数: 0

摘要

对于图 F、G 和 H,让 F→(G,H)表示 F 的任何红/蓝边着色都包含一个红色的 G 或一个蓝色的 H。拉姆齐数 r(G,H) 定义为 Kr→(G,H)的最小整数 r。设 Bn(k) 是由 n 份 Kk+1 组成的书本图,所有 Kk+1 都共享一个共同的 Kk;设 G:=Kp+1(a1,a2,...,ap+1) 是完整的 (p+1) 部分图,其中 a1=1, a2|(n-1) 且 ai≤ai+1 。本文避免使用 Szemerédi 的正则性 Lemma,证明对于任意固定的 p≥1,k≥2 和足够大的 n,Kp(n+a2k-1)+1ȖK1,n+a2-2→(G,Bn(k))。这意味着星形临界拉姆齐数 r⁎(G,Bn(k))=(p-1)(n+a2k-1)+a2(k-1)+1。由此推论,当 a1=a2=1 且 ai≤ai+1 时,r⁎(G,Bn(k))=(p-1)(n+k-1)+k。这就以更强的形式解决了郝和林(2018)[11]提出的问题。
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Star-critical Ramsey numbers involving large books
For graphs F,G and H, let F(G,H) signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number r(G,H) is defined to be the smallest integer r such that Kr(G,H). Let Bn(k) be the book graph which consists of n copies of Kk+1 all sharing a common Kk, and let G:=Kp+1(a1,a2,...,ap+1) be the complete (p+1)-partite graph with a1=1, a2|(n1) and aiai+1.
In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed p1, k2 and sufficiently large n, Kp(n+a2k1)+1K1,n+a22(G,Bn(k)). This implies that the star-critical Ramsey number r(G,Bn(k))=(p1)(n+a2k1)+a2(k1)+1. As a corollary, r(G,Bn(k))=(p1)(n+k1)+k for a1=a2=1 and aiai+1. This solves a problem proposed by Hao and Lin (2018) [11] in a stronger form.
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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