Journal of Combinatorial Theory Series A最新文献

{"title":"Constructing generalized Heffter arrays via near alternating sign matrices","authors":"L. Mella ,&nbsp;T. Traetta","doi":"10.1016/j.jcta.2024.105873","DOIUrl":"10.1016/j.jcta.2024.105873","url":null,"abstract":"<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, th","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139916880","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 maximal number of elements pairwise generating the finite alternating group","authors":"Francesco Fumagalli ,&nbsp;Martino Garonzi ,&nbsp;Pietro Gheri","doi":"10.1016/j.jcta.2024.105870","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105870","url":null,"abstract":"<div><p>Let <em>G</em> be the alternating group of degree <em>n</em>. Let <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the maximal size of a subset <em>S</em> of <em>G</em> such that <span><math><mo>〈</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>〉</mo><mo>=</mo><mi>G</mi></math></span> whenever <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi></math></span> and <span><math><mi>x</mi><mo>≠</mo><mi>y</mi></math></span> and let <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the minimal size of a family of proper subgroups of <em>G</em> whose union is <em>G</em>. We prove that, when <em>n</em> varies in the family of composite numbers, <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> tends to 1 as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, we explicitly calculate <span><math><mi>σ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>21</mn></math></span> congruent to 3 modulo 18.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000098/pdfft?md5=0f713e308f01065a0eed53c25b2ba78c&pid=1-s2.0-S0097316524000098-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139732729","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":"Most plane curves over finite fields are not blocking","authors":"Shamil Asgarli ,&nbsp;Dragos Ghioca ,&nbsp;Chi Hoi Yip","doi":"10.1016/j.jcta.2024.105871","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105871","url":null,"abstract":"<div><p>A plane curve <span><math><mi>C</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of degree <em>d</em> is called <em>blocking</em> if every <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line in the plane meets <em>C</em> at some <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-point. We prove that the proportion of blocking curves among those of degree <em>d</em> is <span><math><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> when <span><math><mi>d</mi><mo>≥</mo><mn>2</mn><mi>q</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition <span><math><mi>d</mi><mo>≥</mo><mn>3</mn><mi>p</mi></math></span> and <span><math><mi>d</mi><mo>,</mo><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-roots of random polynomials, we find that the limiting distribution of the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points in the intersection of a random plane curve and a fixed <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points contained in a union of <em>k</em> lines for <span><math><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139719451","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 Q-polynomial structure for the Attenuated Space poset Aq(N,M)","authors":"Paul Terwilliger","doi":"10.1016/j.jcta.2024.105872","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105872","url":null,"abstract":"<div><p>The goal of this article is to display a <em>Q</em>-polynomial structure for the Attenuated Space poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>. The poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> is briefly described as follows. Start with an <span><math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math></span>-dimensional vector space <em>H</em> over a finite field with <em>q</em> elements. Fix an <em>M</em>-dimensional subspace <em>h</em> of <em>H</em>. The vertex set <em>X</em> of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> consists of the subspaces of <em>H</em> that have zero intersection with <em>h</em>. The partial order on <em>X</em> is the inclusion relation. The <em>Q</em>-polynomial structure involves two matrices <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> with the following entries. For <span><math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math></span> the matrix <em>A</em> has <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math></span>-entry 1 (if <em>y</em> covers <em>z</em>); <span><math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> (if <em>z</em> covers <em>y</em>); and 0 (if neither of <span><math><mi>y</mi><mo>,</mo><mi>z</mi></math></span> covers the other). The matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is diagonal, with <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> for all <span><math><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. By construction, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. By construction, <em>A</em> acts on these eigenspaces in a (block) tridiagonal fashion. We show that <em>A</em> is diagonalizable, with <span><math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. We show that <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that <em>A</em> is <em>Q</em>-polynomial. We show that <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra <em>T</em> of ","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713906","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":"Spectral characterization of the complete graph removing a cycle","authors":"Muhuo Liu ,&nbsp;Xiaofeng Gu ,&nbsp;Haiying Shan ,&nbsp;Zoran Stanić","doi":"10.1016/j.jcta.2024.105868","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105868","url":null,"abstract":"<div><p>A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> vertices by deleting all edges of a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <em>k</em> vertices, is determined by its spectrum for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>, but not for <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span>. In this paper, we show that <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span> is the unique exception for the spectral determination of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713905","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":"The divisor class group of a discrete polymatroid","authors":"Jürgen Herzog ,&nbsp;Takayuki Hibi ,&nbsp;Somayeh Moradi ,&nbsp;Ayesha Asloob Qureshi","doi":"10.1016/j.jcta.2024.105869","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105869","url":null,"abstract":"<div><p>In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139709553","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":"Large sum-free sets in Z5n","authors":"Vsevolod F. Lev","doi":"10.1016/j.jcta.2024.105865","DOIUrl":"10.1016/j.jcta.2024.105865","url":null,"abstract":"<div><p>It is well-known that for a prime <span><math><mi>p</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and integer <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span><span>, the maximum possible size of a sum-free subset of the elementary abelian group </span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mspace></mspace><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. However, the matching stability result is known for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> only. We consider the first open case <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span> showing that if <span><math><mi>A</mi><mo>⊆</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is a sum-free subset with <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>&gt;</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, then there are a subgroup <span><math><mi>H</mi><mo>&lt;</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of size <span><math><mo>|</mo><mi>H</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and an element <span><math><mi>e</mi><mo>∉</mo><mi>H</mi></math></span> such that <span><math><mi>A</mi><mo>⊆</mo><mo>(</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mo>−</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660492","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":"Block-transitive 2-designs with a chain of imprimitive point-partitions","authors":"Carmen Amarra ,&nbsp;Alice Devillers ,&nbsp;Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105866","DOIUrl":"10.1016/j.jcta.2024.105866","url":null,"abstract":"<div><p>More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size <em>k</em>, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of <em>k</em>. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000050/pdfft?md5=4350b32c706696ea2c3f719fb15cd8bc&pid=1-s2.0-S0097316524000050-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660473","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":"Large monochromatic components in colorings of complete hypergraphs","authors":"Lyuben Lichev ,&nbsp;Sammy Luo","doi":"10.1016/j.jcta.2024.105867","DOIUrl":"10.1016/j.jcta.2024.105867","url":null,"abstract":"<div><p>Gyárfás famously showed that in every <em>r</em>-coloring of the edges of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there is a monochromatic connected component with at least <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span> vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for <em>k</em><span>-uniform hypergraphs<span>. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of </span></span><span><math><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> as the number of colors grows. We further generalize these questions to ask about counts of vertex <em>s</em>-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660447","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 study on free roots of Borcherds-Kac-Moody Lie superalgebras","authors":"Shushma Rani ,&nbsp;G. Arunkumar","doi":"10.1016/j.jcta.2024.105862","DOIUrl":"10.1016/j.jcta.2024.105862","url":null,"abstract":"<div><p><span>Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as </span><span><math><mi>g</mi></math></span>, associated with the graph <em>G</em><span>. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph </span><em>G</em>.</p><p><span>The Chevalley relations lead to a triangular decomposition of </span><span><math><mi>g</mi></math></span> as <span><math><mi>g</mi><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>⊕</mo><mi>h</mi><mo>⊕</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span>, where each root space <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is contained in either <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> or <span><math><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub></math></span>. Importantly, each <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is determined solely by relations (2) and (3). This paper focuses on the root spaces of <span><math><mi>g</mi></math></span> that are unaffected by the Serre relations. We refer to these root spaces as “free roots” of <span><math><mi>g</mi></math></span> (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in <span>Lemma 3.10</span>. Consequently, we refer to them as “free roots,” and the corresponding root spaces in <span><math><mi>g</mi></math></span> as “free root spaces” [cf. <span>Definition 2.6</span>]). Since these root spaces only involve commutation relations derived from the graph <em>G</em>, we can examine them purely from a combinatorial perspective.</p><p>We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of <span><math><mi>g</mi></math></span><span>: We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference </span><span>[1]</span>, designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139566035","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}