{"title":"关于有单位分数的方程的拉多数","authors":"Collier Gaiser","doi":"10.1016/j.disc.2024.114156","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> be the smallest positive integer <em>n</em> such that every <em>r</em>-coloring of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> has a monochromatic solution to the nonlinear equation<span><span><span><math><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>/</mo><mi>y</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are not necessarily distinct. Brown and Rödl <span>[3]</span> proved that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>. In this paper, we prove that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>. The main ingredient in our proof is a finite set <span><math><mi>A</mi><mo>⊆</mo><mi>N</mi></math></span> such that every 2-coloring of <em>A</em> has a monochromatic solution to the linear equation <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>y</mi></math></span> and the least common multiple of <em>A</em> is sufficiently small. This approach can also be used to study <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> with <span><math><mi>r</mi><mo>></mo><mn>2</mn></math></span>. For example, a recent result of Boza et al. <span>[2]</span> implies that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>43</mn></mrow></msup><mo>)</mo></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Rado numbers for equations with unit fractions\",\"authors\":\"Collier Gaiser\",\"doi\":\"10.1016/j.disc.2024.114156\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> be the smallest positive integer <em>n</em> such that every <em>r</em>-coloring of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> has a monochromatic solution to the nonlinear equation<span><span><span><math><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>/</mo><mi>y</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are not necessarily distinct. Brown and Rödl <span>[3]</span> proved that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>. In this paper, we prove that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>. The main ingredient in our proof is a finite set <span><math><mi>A</mi><mo>⊆</mo><mi>N</mi></math></span> such that every 2-coloring of <em>A</em> has a monochromatic solution to the linear equation <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>y</mi></math></span> and the least common multiple of <em>A</em> is sufficiently small. This approach can also be used to study <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> with <span><math><mi>r</mi><mo>></mo><mn>2</mn></math></span>. For example, a recent result of Boza et al. <span>[2]</span> implies that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>43</mn></mrow></msup><mo>)</mo></math></span>.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-09\",\"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/S0012365X24002875\",\"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/S0012365X24002875","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
设 fr(k) 是最小的正整数 n,使得{1,2,...,n}的每一个 r 色都有非线性方程 1/x1+⋯+1/xk=1/y 的单色解,其中 x1,...xk 不一定是不同的。Brown 和 Rödl [3] 证明了 f2(k)=O(k6) 。本文将证明 f2(k)=O(k3)。我们证明的主要要素是一个有限集 A⊆N,使得 A 的每个 2 色都有线性方程 x1+⋯+xk=y 的单色解,并且 A 的最小公倍数足够小。例如,Boza 等人[2]的最新结果暗示 f3(k)=O(k43)。
Let be the smallest positive integer n such that every r-coloring of has a monochromatic solution to the nonlinear equation where are not necessarily distinct. Brown and Rödl [3] proved that . In this paper, we prove that . The main ingredient in our proof is a finite set such that every 2-coloring of A has a monochromatic solution to the linear equation and the least common multiple of A is sufficiently small. This approach can also be used to study with . For example, a recent result of Boza et al. [2] implies that .
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.