CombinatoricaPub Date : 2025-01-02DOI: 10.1007/s00493-024-00125-z
Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao
{"title":"Uniacute Spherical Codes","authors":"Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao","doi":"10.1007/s00493-024-00125-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00125-z","url":null,"abstract":"<p>A spherical <i>L</i>-code, where <span>(L subseteq [-1,infty ))</span>, consists of unit vectors in <span>(mathbb {R}^d)</span> whose pairwise inner products are contained in <i>L</i>. Determining the maximum cardinality <span>(N_L(d))</span> of an <i>L</i>-code in <span>(mathbb {R}^d)</span> is a fundamental question in discrete geometry and has been extensively investigated for various choices of <i>L</i>. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to <span>(L = {-alpha , alpha })</span>, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that <span>(N_L(d) = O_L(d))</span> for <span>(L = [-1, -beta ] cup {alpha })</span> with <span>(alpha ,beta > 0)</span> (we call such <i>L</i>-codes “uniacute”), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a “uniform bound” showing <span>(limsup _{drightarrow infty } N_L(d)/d le 2p)</span> for <span>(L = [-1, -beta ] cup {alpha })</span> and <span>(p = lfloor alpha /beta rfloor + 1)</span>. For which <span>((alpha ,beta ))</span> is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate <i>p</i>-block structure. We also formulate a notion of “modular codes,” which we conjecture to be optimal in high dimensions.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"375 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142916856","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}
CombinatoricaPub Date : 2024-12-29DOI: 10.1007/s00493-024-00129-9
Agelos Georgakopoulos
{"title":"The Excluded Minors for Embeddability into a Compact Surface","authors":"Agelos Georgakopoulos","doi":"10.1007/s00493-024-00129-9","DOIUrl":"https://doi.org/10.1007/s00493-024-00129-9","url":null,"abstract":"<p>We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142887799","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}
CombinatoricaPub Date : 2024-12-29DOI: 10.1007/s00493-024-00132-0
Antonio Montero, Micael Toledo
{"title":"Chiral Extensions of Regular Toroids","authors":"Antonio Montero, Micael Toledo","doi":"10.1007/s00493-024-00132-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00132-0","url":null,"abstract":"<p>Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the <i>n</i>-dimensional torus (<span>(n geqslant 2)</span>). As a consequence, we prove that for every <span>(d geqslant 3)</span> there exist infinitely many chiral <i>d</i>-polytopes.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"153 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142887797","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}
CombinatoricaPub Date : 2024-12-20DOI: 10.1007/s00493-024-00133-z
Xiao Han
{"title":"A New Bound for the Fourier-Entropy-Influence Conjecture","authors":"Xiao Han","doi":"10.1007/s00493-024-00133-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00133-z","url":null,"abstract":"<p>In this paper, we prove that the Fourier entropy of an <i>n</i>-dimensional boolean function <i>f</i> can be upper-bounded by <span>(O(I(f)+ sum limits _{kin [n]}I_k(f)log frac{1}{I_k(f)}))</span>, where <i>I</i>(<i>f</i>) is its total influence and <span>(I_k(f))</span> is the influence of the <i>k</i>-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture <span>(O(I(f)log I(f)))</span> and <span>(O(I(f)^2))</span>. The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels to estimate the Fourier entropy as its derivative.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"69 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142939884","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}
CombinatoricaPub Date : 2024-12-18DOI: 10.1007/s00493-024-00131-1
Rafał Kalinowski, Monika Pilśniak, Marcin Stawiski
{"title":"Unfriendly Partition Conjecture Holds for Line Graphs","authors":"Rafał Kalinowski, Monika Pilśniak, Marcin Stawiski","doi":"10.1007/s00493-024-00131-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00131-1","url":null,"abstract":"<p>A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex <i>v</i> and every color <span>(alpha )</span>, there are at most as many edges incident to <i>v</i> colored with <span>(alpha )</span> as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges has a majority edge-coloring from lists of size 4. Another interesting result states that every infinite graph without vertices of finite odd degrees admits a majority edge-coloring from lists of size 2. As a consequence of our results, we prove that line graphs of any cardinality admit majority vertex-colorings from lists of size 2, thus confirming the Unfriendly Partition Conjecture for line graphs.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"82 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841446","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}
CombinatoricaPub Date : 2024-12-18DOI: 10.1007/s00493-024-00130-2
Siddharth Bhandari, Abhishek Khetan
{"title":"Improved Upper Bound for the Size of a Trifferent Code","authors":"Siddharth Bhandari, Abhishek Khetan","doi":"10.1007/s00493-024-00130-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00130-2","url":null,"abstract":"<p>A subset <span>(mathcal {C}subseteq {0,1,2}^n)</span> is said to be a <i>trifferent</i> code (of block length <i>n</i>) if for every three distinct codewords <span>(x,y, z in mathcal {C})</span>, there is a coordinate <span>(iin {1,2,ldots ,n})</span> where they all differ, that is, <span>({x(i),y(i),z(i)})</span> is same as <span>({0,1,2})</span>. Let <i>T</i>(<i>n</i>) denote the size of the largest trifferent code of block length <i>n</i>. Understanding the asymptotic behavior of <i>T</i>(<i>n</i>) is closely related to determining the zero-error capacity of the (3/2)-channel defined by Elias (IEEE Trans Inform Theory 34(5):1070–1074, 1988), and is a long-standing open problem in the area. Elias had shown that <span>(T(n)le 2times (3/2)^n)</span> and prior to our work the best upper bound was <span>(T(n)le 0.6937 times (3/2)^n)</span> due to Kurz (Example Counterexample 5:100139, 2024). We improve this bound to <span>(T(n)le c times n^{-2/5}times (3/2)^n)</span> where <i>c</i> is an absolute constant.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"36 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841564","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}
CombinatoricaPub Date : 2024-12-17DOI: 10.1007/s00493-024-00124-0
Tomas Juškevičius, Valentas Kurauskas
{"title":"Anticoncentration of Random Vectors via the Strong Perfect Graph Theorem","authors":"Tomas Juškevičius, Valentas Kurauskas","doi":"10.1007/s00493-024-00124-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00124-0","url":null,"abstract":"<p>In this paper we give anticoncentration bounds for sums of independent random vectors in finite-dimensional vector spaces. In particular, we asymptotically establish a conjecture of Leader and Radcliffe (SIAM J Discrete Math 7:90–101, 1994) and a question of Jones (SIAM J Appl Math 34:1–6, 1978). The highlight of this work is an application of the strong perfect graph theorem by Chudnovsky et al. (Ann Math 164:51–229, 2006) in the context of anticoncentration.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"39 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142832150","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}
CombinatoricaPub Date : 2024-11-07DOI: 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":"62 1","pages":""},"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}
CombinatoricaPub Date : 2024-08-15DOI: 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":"14 1","pages":""},"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}
CombinatoricaPub Date : 2024-08-05DOI: 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<c<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>c)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"18 1","pages":""},"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}
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