{"title":"Online size Ramsey numbers: Path vs C4","authors":"","doi":"10.1016/j.disc.2024.114214","DOIUrl":null,"url":null,"abstract":"<div><p>Given two graphs <em>G</em> and <em>H</em>, a size Ramsey game is played on the edge set of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span>. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of <em>G</em> or a blue copy of <em>H</em> as soon as possible. The online (size) Ramsey number <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the number of rounds in the game provided Builder and Painter play optimally. We prove that <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. Our proof for <span><math><mi>n</mi><mo>≤</mo><mn>13</mn></math></span> is computer-assisted. The bound <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> solves also the “all cycles vs. <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>” game for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span> – it implies that it takes Builder <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> rounds to force Painter to create a blue path on <em>n</em> vertices or any red cycle.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-08-20","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/S0012365X24003455","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
Abstract
Given two graphs G and H, a size Ramsey game is played on the edge set of . In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of G or a blue copy of H as soon as possible. The online (size) Ramsey number is the number of rounds in the game provided Builder and Painter play optimally. We prove that for every . The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get for . Our proof for is computer-assisted. The bound solves also the “all cycles vs. ” game for – it implies that it takes Builder rounds to force Painter to create a blue path on n vertices or any red cycle.
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.