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Any Two-Coloring of the Plane Contains Monochromatic 3-Term Arithmetic Progressions 平面的任意双色包含单色三项算术级数
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-11-07 DOI: 10.1007/s00493-024-00122-2
Gabriel Currier, Kenneth Moore, Chi Hoi Yip
{"title":"Any Two-Coloring of the Plane Contains Monochromatic 3-Term Arithmetic Progressions","authors":"Gabriel Currier, Kenneth Moore, Chi Hoi Yip","doi":"10.1007/s00493-024-00122-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00122-2","url":null,"abstract":"<p>A conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point configuration. This conjecture is known only for special classes of configurations. In this manuscript, we confirm one of the most natural open cases; that is, every two-coloring of the plane admits a monochromatic congruent copy of any 3-term arithmetic progression.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hamilton Transversals in Tournaments 锦标赛中的汉密尔顿横轴
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-08-15 DOI: 10.1007/s00493-024-00123-1
Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo
{"title":"Hamilton Transversals in Tournaments","authors":"Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo","doi":"10.1007/s00493-024-00123-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00123-1","url":null,"abstract":"<p>It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes <i>transversal</i> generalizations of these classical results. For a collection <span>(textbf{T}=(T_1,dots ,T_m))</span> of not-necessarily distinct tournaments on a common vertex set <i>V</i>, an <i>m</i>-edge directed graph <span>(mathcal {D})</span> with vertices in <i>V</i> is called a <span>(textbf{T})</span>-transversal if there exists a bijection <span>(phi :E(mathcal {D})rightarrow [m])</span> such that <span>(ein E(T_{phi (e)}))</span> for all <span>(ein E(mathcal {D}))</span>. We prove that for sufficiently large <i>m</i> with <span>(m=|V|-1)</span>, there exists a <span>(textbf{T})</span>-transversal Hamilton path. Moreover, if <span>(m=|V|)</span> and at least <span>(m-1)</span> of the tournaments <span>(T_1,ldots ,T_m)</span> are assumed to be strongly connected, then there is a <span>(textbf{T})</span>-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub <span>(textbf{H})</span>-<i>partition</i>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141986586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Pure Pairs. VIII. Excluding a Sparse Graph Pure Pairs.VIII.排除稀疏图
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-08-05 DOI: 10.1007/s00493-024-00117-z
Alex Scott, Paul Seymour, Sophie Spirkl
{"title":"Pure Pairs. VIII. Excluding a Sparse Graph","authors":"Alex Scott, Paul Seymour, Sophie Spirkl","doi":"10.1007/s00493-024-00117-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00117-z","url":null,"abstract":"<p>A pure pair of size <i>t</i> in a graph <i>G</i> is a pair <i>A</i>, <i>B</i> of disjoint subsets of <i>V</i>(<i>G</i>), each of cardinality at least <i>t</i>, such that <i>A</i> is either complete or anticomplete to <i>B</i>. It is known that, for every forest <i>H</i>, every graph on <span>(nge 2)</span> vertices that does not contain <i>H</i> or its complement as an induced subgraph has a pure pair of size <span>(Omega (n))</span>; furthermore, this only holds when <i>H</i> or its complement is a forest. In this paper, we look at pure pairs of size <span>(n^{1-c})</span>, where <span>(0&lt;c&lt;1)</span>. Let <i>H</i> be a graph: does every graph on <span>(nge 2)</span> vertices that does not contain <i>H</i> or its complement as an induced subgraph have a pure pair of size <span>(Omega (|G|^{1-c}))</span>? The answer is related to the <i>congestion</i> of <i>H</i>, the maximum of <span>(1-(|J|-1)/|E(J)|)</span> over all subgraphs <i>J</i> of <i>H</i> with an edge. (Congestion is nonnegative, and equals zero exactly when <i>H</i> is a forest.) Let <i>d</i> be the smaller of the congestions of <i>H</i> and <span>(overline{H})</span>. We show that the answer to the question above is “yes” if <span>(dle c/(9+15c))</span>, and “no” if <span>(d&gt;c)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141891847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Perfect Matchings in Random Sparsifications of Dirac Hypergraphs 狄拉克超图随机稀疏化中的完美匹配
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-08-05 DOI: 10.1007/s00493-024-00116-0
Dong Yeap Kang, Tom Kelly, Daniela Kühn, Deryk Osthus, Vincent Pfenninger
{"title":"Perfect Matchings in Random Sparsifications of Dirac Hypergraphs","authors":"Dong Yeap Kang, Tom Kelly, Daniela Kühn, Deryk Osthus, Vincent Pfenninger","doi":"10.1007/s00493-024-00116-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00116-0","url":null,"abstract":"<p>For all integers <span>(n ge k &gt; d ge 1)</span>, let <span>(m_{d}(k,n))</span> be the minimum integer <span>(D ge 0)</span> such that every <i>k</i>-uniform <i>n</i>-vertex hypergraph <span>({mathcal {H}})</span> with minimum <i>d</i>-degree <span>(delta _{d}({mathcal {H}}))</span> at least <i>D</i> has an optimal matching. For every fixed integer <span>(k ge 3)</span>, we show that for <span>(n in k mathbb {N})</span> and <span>(p = Omega (n^{-k+1} log n))</span>, if <span>({mathcal {H}})</span> is an <i>n</i>-vertex <i>k</i>-uniform hypergraph with <span>(delta _{k-1}({mathcal {H}}) ge m_{k-1}(k,n))</span>, then a.a.s. its <i>p</i>-random subhypergraph <span>({mathcal {H}}_p)</span> contains a perfect matching. Moreover, for every fixed integer <span>(d &lt; k)</span> and <span>(gamma &gt; 0)</span>, we show that the same conclusion holds if <span>({mathcal {H}})</span> is an <i>n</i>-vertex <i>k</i>-uniform hypergraph with <span>(delta _d({mathcal {H}}) ge m_{d}(k,n) + gamma left( {begin{array}{c}n - d k - dend{array}}right) )</span>. Both of these results strengthen Johansson, Kahn, and Vu’s seminal solution to Shamir’s problem and can be viewed as “robust” versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, <span>({mathcal {H}})</span> has at least <span>(exp ((1-1/k)n log n - Theta (n)))</span> many perfect matchings, which is best possible up to an <span>(exp (Theta (n)))</span> factor.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141891849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Whitney Type Theorem for Surfaces: Characterising Graphs with Locally Planar Embeddings 曲面的惠特尼型定理：用局部平面嵌入描述图形的特征
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-23 DOI: 10.1007/s00493-024-00118-y
Johannes Carmesin
{"title":"A Whitney Type Theorem for Surfaces: Characterising Graphs with Locally Planar Embeddings","authors":"Johannes Carmesin","doi":"10.1007/s00493-024-00118-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00118-y","url":null,"abstract":"<p>Given a graph <i>G</i> and a parameter <i>r</i>, we define the <i>r</i>-<i>local matroid</i> of <i>G</i> to be the matroid generated by its cycles of length at most <i>r</i>. Extending Whitney’s abstract planar duality theorem from 1932, we prove that for every <i>r</i> the <i>r</i>-local matroid of <i>G</i> is co-graphic if and only if <i>G</i> admits a certain type of embedding in a surface, which we call <i>r</i>-<i>planar embedding</i>. The maximum value of <i>r</i> such that a graph <i>G</i> admits an <i>r</i>-planar embedding is closely related to face-width, and such embeddings for this maximum value of <i>r</i> are quite often embeddings of minimum genus. Unlike minimum genus embeddings, these <i>r</i>-planar embeddings can be computed in polynomial time. This provides the first systematic and polynomially computable method to construct for every graph <i>G</i> a surface so that <i>G</i> embeds in that surface in an optimal way (phrased in our notion of <i>r</i>-planarity).</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Storage Codes on Coset Graphs with Asymptotically Unit Rate 具有渐近单位速率的余集图存储代码
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-23 DOI: 10.1007/s00493-024-00114-2
Alexander Barg, Moshe Schwartz, Lev Yohananov
{"title":"Storage Codes on Coset Graphs with Asymptotically Unit Rate","authors":"Alexander Barg, Moshe Schwartz, Lev Yohananov","doi":"10.1007/s00493-024-00114-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00114-2","url":null,"abstract":"<p>A storage code on a graph <i>G</i> is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., Cameron et al., in: Electron J Combin 23(1):48, 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unavoidable Flats in Matroids Representable over Prime Fields 可在素域上表示的矩阵中不可避免的平面
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-11 DOI: 10.1007/s00493-024-00112-4
Jim Geelen, Matthew E. Kroeker
{"title":"Unavoidable Flats in Matroids Representable over Prime Fields","authors":"Jim Geelen, Matthew E. Kroeker","doi":"10.1007/s00493-024-00112-4","DOIUrl":"https://doi.org/10.1007/s00493-024-00112-4","url":null,"abstract":"<p>We show that, for any prime <i>p</i> and integer <span>(k ge 2)</span>, a simple <span>({{,textrm{GF},}}(p))</span>-representable matroid with sufficiently high rank has a rank-<i>k</i> flat which is either independent in <i>M</i>, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for <span>({{,textrm{GF},}}(p))</span>-representable matroids. For any prime <i>p</i> and integer <span>(kge 2)</span>, if we 2-colour the elements in any simple <span>({{,textrm{GF},}}(p))</span>-representable matroid with sufficiently high rank, then there is a monochromatic flat of rank <i>k</i>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Pisier Type Theorems 论皮西埃类型定理
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-11 DOI: 10.1007/s00493-024-00115-1
Jaroslav Nešetřil, Vojtěch Rödl, Marcelo Sales
{"title":"On Pisier Type Theorems","authors":"Jaroslav Nešetřil, Vojtěch Rödl, Marcelo Sales","doi":"10.1007/s00493-024-00115-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00115-1","url":null,"abstract":"<p>For any integer <span>(hgeqslant 2)</span>, a set of integers <span>(B={b_i}_{iin I})</span> is a <span>(B_h)</span>-set if all <i>h</i>-sums <span>(b_{i_1}+ldots +b_{i_h})</span> with <span>(i_1&lt;ldots &lt;i_h)</span> are distinct. Answering a question of Alon and Erdős [2], for every <span>(hgeqslant 2)</span> we construct a set of integers <i>X</i> which is not a union of finitely many <span>(B_h)</span>-sets, yet any finite subset <span>(Ysubseteq X)</span> contains an <span>(B_h)</span>-set <i>Z</i> with <span>(|Z|geqslant varepsilon |Y|)</span>, where <span>(varepsilon :=varepsilon (h))</span>. We also discuss questions related to a problem of Pisier about the existence of a set <i>A</i> with similar properties when replacing <span>(B_h)</span>-sets by the requirement that all finite sums <span>(sum _{jin J}b_j)</span> are distinct.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Reconstruction in One Dimension from Unlabeled Euclidean Lengths 从无标注的欧氏长度重建一维图像
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-11 DOI: 10.1007/s00493-024-00119-x
Robert Connelly, Steven J. Gortler, Louis Theran
{"title":"Reconstruction in One Dimension from Unlabeled Euclidean Lengths","authors":"Robert Connelly, Steven J. Gortler, Louis Theran","doi":"10.1007/s00493-024-00119-x","DOIUrl":"https://doi.org/10.1007/s00493-024-00119-x","url":null,"abstract":"<p>Let <i>G</i> be a 3-connected ordered graph with <i>n</i> vertices and <i>m</i> edges. Let <span>(textbf{p})</span> be a randomly chosen mapping of these <i>n</i> vertices to the integer range <span>({1, 2,3, ldots , 2^b})</span> for <span>(bge m^2)</span>. Let <span>(ell )</span> be the vector of <i>m</i> Euclidean lengths of <i>G</i>’s edges under <span>(textbf{p})</span>. In this paper, we show that, with high probability over <span>(textbf{p})</span>, we can efficiently reconstruct both <i>G</i> and <span>(textbf{p})</span> from <span>(ell )</span>. This reconstruction problem is NP-HARD in the worst case, even if both <i>G</i> and <span>(ell )</span> are given. We also show that our results stand in the presence of small amounts of error in <span>(ell )</span>, and in the real setting, with sufficiently accurate length measurements. Our method combines lattice reduction, which has previously been used to solve random subset sum problems, with an algorithm of Seymour that can efficiently reconstruct an ordered graph given an independence oracle for its matroid.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Directed and Undirected Diameters of Vertex-Transitive Graphs 论顶点变换图的有向和无向直径
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-07-09 DOI: 10.1007/s00493-024-00120-4
Saveliy V. Skresanov
{"title":"On Directed and Undirected Diameters of Vertex-Transitive Graphs","authors":"Saveliy V. Skresanov","doi":"10.1007/s00493-024-00120-4","DOIUrl":"https://doi.org/10.1007/s00493-024-00120-4","url":null,"abstract":"<p>A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the edges. In 2006 Babai proved that for a connected directed Cayley graph on <span>( n )</span> vertices the directed diameter is bounded above by a polynomial in undirected diameter and <span>( log n )</span>. Moreover, Babai conjectured that a similar bound holds for vertex-transitive graphs. We prove this conjecture of Babai, in fact, it follows from a more general bound for connected relations of homogeneous coherent configurations. The main novelty of the proof is a generalization of Ruzsa’s triangle inequality from additive combinatorics to the setting of graphs</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141561508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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