{"title":"Fan-complete Ramsey numbers","authors":"Fan Chung , Qizhong Lin","doi":"10.1016/j.aam.2025.102939","DOIUrl":"10.1016/j.aam.2025.102939","url":null,"abstract":"<div><div>For graphs <em>G</em> and <em>H</em>, we consider Ramsey numbers <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> with tight lower bounds, namely, <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the chromatic number of <em>G</em> and <span><math><mo>|</mo><mi>H</mi><mo>|</mo></math></span> denotes the number of vertices in <em>H</em>. We say <em>H</em> is <em>G</em>-good if the equality holds.</div><div>Let <span><math><mi>G</mi><mo>+</mo><mi>H</mi></math></span> be the join graph obtained from graphs <em>G</em> and <em>H</em> by adding all edges between the disjoint vertex sets of <em>G</em> and <em>H</em>. Let <em>nH</em> denote the union graph of <em>n</em> disjoint copies of <em>H</em>. We show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><mi>H</mi></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <em>n</em> is sufficiently large. In particular, the fan-graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <span><math><mi>n</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, improving previous tower-type lower bounds for <em>n</em> due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>F</mi><mo>)</mo></math></span> for the case of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>, the complete <em>p</em>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph <em>H</em>,<span><span><span><math><mrow><mi>r</mi><mo>(</mo><mi>G</m","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102939"},"PeriodicalIF":1.0,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The number of edges in graphs with bounded clique number and circumference","authors":"Chunyang Dou , Bo Ning , Xing Peng","doi":"10.1016/j.aam.2025.102936","DOIUrl":"10.1016/j.aam.2025.102936","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the maximum possible number of edges in an <em>n</em>-vertex graph which does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a matching of size <em>k</em>. Replacing <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, Katona and Xiao obtained the Turán number of <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> for <span><math><mi>r</mi><mo>≤</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> and sufficiently large <em>n</em>. In addition, they proposed a conjecture for the case where <span><math><mi>r</mi><mo>≥</mo><mo>⌊</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span> and <em>n</em> is sufficiently large. Motivated by the fact that the result for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> can be deduced from the one for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, we investigate the Turán number of <span><math><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span> in this paper, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math></span> denotes the set of cycles of length at least <em>k</em>. In other words, we aim to determine the maximum number of edges in graphs with clique number at most <span><math><mi>r</mi><mo>−</mo><mn>1</mn></math></span> and circumference at most <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. For <span><math><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>}</mo></math></span>, we are able to show the value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102936"},"PeriodicalIF":1.0,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A relationship for LYM inequalities between Boolean lattices and linear lattices with applications","authors":"Jiuqiang Liu , Guihai Yu","doi":"10.1016/j.aam.2025.102935","DOIUrl":"10.1016/j.aam.2025.102935","url":null,"abstract":"<div><div>Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no <em>s</em> pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no <em>d</em>-dimensional Boolean algebras.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102935"},"PeriodicalIF":1.0,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144613816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the rigidity of Arnoux-Rauzy words","authors":"V. Berthé , S. Puzynina","doi":"10.1016/j.aam.2025.102932","DOIUrl":"10.1016/j.aam.2025.102932","url":null,"abstract":"<div><div>An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102932"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values","authors":"S. Yakubovich","doi":"10.1016/j.aam.2025.102934","DOIUrl":"10.1016/j.aam.2025.102934","url":null,"abstract":"<div><div>We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102934"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak maps and the Tutte polynomial","authors":"Christine Cho, James Oxley","doi":"10.1016/j.aam.2025.102933","DOIUrl":"10.1016/j.aam.2025.102933","url":null,"abstract":"<div><div>Let <em>M</em> and <em>N</em> be matroids such that <em>N</em> is the image of <em>M</em> under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for <em>x</em> and <em>y</em> positive, <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>≥</mo><mi>T</mi><mo>(</mo><mi>N</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> if and only if <span><math><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mi>x</mi><mi>y</mi></math></span> or <span><math><mi>M</mi><mo>≅</mo><mi>N</mi></math></span>. We give a number of consequences of this result.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102933"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A polynomial time algorithm for Sylvester waves when entries are bounded","authors":"Guoce Xin , Chen Zhang","doi":"10.1016/j.aam.2025.102931","DOIUrl":"10.1016/j.aam.2025.102931","url":null,"abstract":"<div><div>Sylvester's denumerant <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> is a quantity that counts the number of nonnegative integer solutions to the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>t</mi></math></span>, where <span><math><mi>a</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>)</mo></math></span> is a sequence of positive integers with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We present a polynomial time algorithm in <em>N</em> for computing <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> when <strong><em>a</em></strong> is bounded and <em>t</em> is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in <span>Maple</span> under the name <span>Cyc-Denum</span> and demonstrates superior performance when <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>500</mn></math></span> compared to Sills-Zeilberger's <span>Maple</span> package <span>PARTITIONS</span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102931"},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144535520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann
{"title":"Algebraic geometry of quantum graphical models","authors":"Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann","doi":"10.1016/j.aam.2025.102930","DOIUrl":"10.1016/j.aam.2025.102930","url":null,"abstract":"<div><div>Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102930"},"PeriodicalIF":1.0,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Classifying one-dimensional discrete models with maximum likelihood degree one","authors":"Arthur Bik , Orlando Marigliano","doi":"10.1016/j.aam.2025.102928","DOIUrl":"10.1016/j.aam.2025.102928","url":null,"abstract":"<div><div>We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102928"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144330513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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