{"title":"路径的对角线外在线拉姆齐数的渐近线","authors":"","doi":"10.1016/j.ejc.2024.104032","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that for every <span><math><mrow><mi>k</mi><mo>≥</mo><mn>10</mn></mrow></math></span>, the online Ramsey number for paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfies <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>9</mn></mrow></mfrac><mo>−</mo><mn>4</mn></mrow></math></span>, matching up to a linear term in <span><math><mi>k</mi></math></span> the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span>, whenever <span><math><mrow><mn>10</mn><mo>≤</mo><mi>k</mi><mo>=</mo><mi>o</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, disproving a conjecture by Cyman et al. (2015).</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001173/pdfft?md5=649b71645f651f52b84e0b2428cf8265&pid=1-s2.0-S0195669824001173-main.pdf","citationCount":"0","resultStr":"{\"title\":\"The asymptotic of off-diagonal online Ramsey numbers for paths\",\"authors\":\"\",\"doi\":\"10.1016/j.ejc.2024.104032\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that for every <span><math><mrow><mi>k</mi><mo>≥</mo><mn>10</mn></mrow></math></span>, the online Ramsey number for paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfies <span><math><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>9</mn></mrow></mfrac><mo>−</mo><mn>4</mn></mrow></math></span>, matching up to a linear term in <span><math><mi>k</mi></math></span> the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span>, whenever <span><math><mrow><mn>10</mn><mo>≤</mo><mi>k</mi><mo>=</mo><mi>o</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, disproving a conjecture by Cyman et al. (2015).</p></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001173/pdfft?md5=649b71645f651f52b84e0b2428cf8265&pid=1-s2.0-S0195669824001173-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/S0195669824001173\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001173","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The asymptotic of off-diagonal online Ramsey numbers for paths
We prove that for every , the online Ramsey number for paths and satisfies , matching up to a linear term in the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies , whenever , disproving a conjecture by Cyman et al. (2015).
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.
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