Journal of Combinatorial Theory Series A最新文献

{"title":"Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)","authors":"Yasutsugu Fujita , Maohua Le","doi":"10.1016/j.jcta.2024.106004","DOIUrl":"10.1016/j.jcta.2024.106004","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span>, <span><math><mi>P</mi></math></span> be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma <span><span>[14]</span></span> conjectured that the equation <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>p</mi><mo>∈</mo><mi>P</mi><mo>,</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> has only one solution <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>49</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is a solution of <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> with <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, then there exist an odd positive integer <em>g</em> and a positive integer <em>t</em> which make <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi></mrow></msup><mo>=</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mo>(</mo><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>/</mo><mo>(</mo><mi>α</mi><mo>−</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> for any integer <em>r</em>, <span><math><mi>α</mi><mo>=</mo><mn>2</mn><mi>g</mi><mo>+</mo><msqrt><mrow><mn>4</mn><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106004"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100974","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}

{"title":"Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions","authors":"Gábor P. Nagy","doi":"10.1016/j.jcta.2024.106001","DOIUrl":"10.1016/j.jcta.2024.106001","url":null,"abstract":"<div><div>The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, there should be Sidon sets of size at least <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> for all <em>n</em>. This paper provides an overview of the known large Sidon sets in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106001"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874013","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}

{"title":"Diametric problem for permutations with the Ulam metric (optimal anticodes)","authors":"Pat Devlin,&nbsp;Leo Douhovnikoff","doi":"10.1016/j.jcta.2024.106002","DOIUrl":"10.1016/j.jcta.2024.106002","url":null,"abstract":"<div><div>We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of permutations on <em>n</em> symbols, and for each <span><math><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most <em>k</em> has size at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>+</mo><mi>C</mi><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></msup><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>, compared to the best known construction of size <span><math><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106002"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095724","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}

{"title":"Cayley extensions of maniplexes and polytopes","authors":"Gabe Cunningham ,&nbsp;Elías Mochán ,&nbsp;Antonio Montero","doi":"10.1016/j.jcta.2024.106000","DOIUrl":"10.1016/j.jcta.2024.106000","url":null,"abstract":"<div><div>A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define <span><math><mi>M</mi></math></span> to be a <em>Cayley extension</em> of <span><math><mi>K</mi></math></span> if the facets of <span><math><mi>M</mi></math></span> are isomorphic to <span><math><mi>K</mi></math></span> and if some subgroup of the automorphism group of <span><math><mi>M</mi></math></span> acts regularly on the facets of <span><math><mi>M</mi></math></span>. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106000"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095348","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}

{"title":"On the size of integer programs with bounded non-vanishing subdeterminants","authors":"Björn Kriepke,&nbsp;Gohar M. Kyureghyan,&nbsp;Matthias Schymura","doi":"10.1016/j.jcta.2024.106003","DOIUrl":"10.1016/j.jcta.2024.106003","url":null,"abstract":"<div><div>Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank <em>r</em> and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> minor is non-zero and bounded by a fixed constant Δ in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span>, and a discussion of a computational approach to determine the maximal number of columns for small parameters Δ and <em>r</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106003"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On joint short minimal zero-sum subsequences over finite abelian groups of rank two","authors":"Yushuang Fan ,&nbsp;Qinghai Zhong","doi":"10.1016/j.jcta.2024.105984","DOIUrl":"10.1016/j.jcta.2024.105984","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> be a finite abelian group and let <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest integer <em>ℓ</em> such that every sequence over <span><math><mi>G</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> of length <em>ℓ</em> has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and solved the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> is a prime. In this paper, we determine the precise value of <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, which confirms a conjecture of Gao, Geroldinger and Wang for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> except <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105984"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The degree of functions in the Johnson and q-Johnson schemes","authors":"Michael Kiermaier ,&nbsp;Jonathan Mannaert ,&nbsp;Alfred Wassermann","doi":"10.1016/j.jcta.2024.105979","DOIUrl":"10.1016/j.jcta.2024.105979","url":null,"abstract":"<div><div>In 1982, Cameron and Liebler investigated certain <em>special sets of lines</em> in <span><math><mi>PG</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these <em>Cameron-Liebler line classes</em> got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the <em>Johnson</em> and <em>q-Johnson schemes</em>.</div><div>In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of <em>designs</em> and <em>antidesigns</em> in association schemes. This leads to a natural definition of the <em>degree</em> and the <em>weights</em> of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree <em>t</em> will be proven.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105979"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746446","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}

{"title":"Sequence reconstruction problem for deletion channels: A complete asymptotic solution","authors":"Van Long Phuoc Pham ,&nbsp;Keshav Goyal ,&nbsp;Han Mao Kiah","doi":"10.1016/j.jcta.2024.105980","DOIUrl":"10.1016/j.jcta.2024.105980","url":null,"abstract":"<div><div>Transmit a codeword <figure><img></figure>, that belongs to an <span><math><mo>(</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-deletion-correcting code of length <em>n</em>, over a <em>t</em>-deletion channel for some <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>t</mi><mo>&lt;</mo><mi>n</mi></math></span>. Levenshtein (2001) <span><span>[10]</span></span>, proposed the problem of determining <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, the minimum number of distinct channel outputs required to uniquely reconstruct <figure><img></figure>. Prior to this work, <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is known only when <span><math><mi>ℓ</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span>. Here, we provide an asymptotically exact solution for all values of <em>ℓ</em> and <em>t</em>. Specifically, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi></mrow></msup><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>. In the special instances: where <span><math><mi>ℓ</mi><mo>=</mo><mi>t</mi></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>; and when <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>t</mi><mo>=</mo><mn>4</mn></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>≤</mo><mn>20</mn><mi>n</mi><mo>−</mo><mn>150</mn></math></span>. We also provide a conjecture on the exact value of <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for all values of <em>n</em>, <em>ℓ</em>, and <em>t</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105980"},"PeriodicalIF":0.9,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720012","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}

{"title":"Non-empty pairwise cross-intersecting families","authors":"Yang Huang,&nbsp;Yuejian Peng","doi":"10.1016/j.jcta.2024.105981","DOIUrl":"10.1016/j.jcta.2024.105981","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Two families &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting if &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We call &lt;em&gt;t&lt;/em&gt; families &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; pairwise cross-intersecting families if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Additionally, if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then we say that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are non-empty pairwise cross-intersecting. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be non-empty pairwise cross-intersecting families with &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105981"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720004","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}

{"title":"A classification of the flag-transitive 2-(v,k,2) designs","authors":"Hongxue Liang ,&nbsp;Alessandro Montinaro","doi":"10.1016/j.jcta.2024.105983","DOIUrl":"10.1016/j.jcta.2024.105983","url":null,"abstract":"<div><div>In this paper, we provide a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as <em>t</em>-spreads, translation planes, quadrics and Segre varieties.</div><div>Our result together with those of Alavi et al. <span><span>[1]</span></span>, <span><span>[2]</span></span>, Praeger et al. <span><span>[17]</span></span>, Zhou and the first author <span><span>[39]</span></span>, <span><span>[40]</span></span> provides a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105983"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}