Patrick Bennett , Ryan Cushman , Andrzej Dudek , Paweł Prałat
{"title":"Erdős-Gyárfás 函数 f(n,4,5)=56n+o(n) - 所以 Gyárfás 是对的","authors":"Patrick Bennett , Ryan Cushman , Andrzej Dudek , Paweł Prałat","doi":"10.1016/j.jctb.2024.07.001","DOIUrl":null,"url":null,"abstract":"<div><p>A <span><math><mo>(</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span>-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> where every 4-clique spans at least five colors. We show that there exist <span><math><mo>(</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span>-colorings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> using <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Erdős-Gyárfás function f(n,4,5)=56n+o(n) — So Gyárfás was right\",\"authors\":\"Patrick Bennett , Ryan Cushman , Andrzej Dudek , Paweł Prałat\",\"doi\":\"10.1016/j.jctb.2024.07.001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A <span><math><mo>(</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span>-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> where every 4-clique spans at least five colors. We show that there exist <span><math><mo>(</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span>-colorings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> using <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mi>n</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.</p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895624000601\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895624000601","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
The Erdős-Gyárfás function f(n,4,5)=56n+o(n) — So Gyárfás was right
A -coloring of is an edge-coloring of where every 4-clique spans at least five colors. We show that there exist -colorings of using colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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