Pub Date : 2026-01-28DOI: 10.1007/s00493-025-00194-8
Matthew Kwan, Roodabeh Safavi, Yiting Wang
One of the foundational theorems of extremal graph theory is <jats:italic>Dirac’s theorem</jats:italic> , which says that if an <jats:italic>n</jats:italic> -vertex graph <jats:italic>G</jats:italic> has minimum degree at least <jats:italic>n</jats:italic> /2, then <jats:italic>G</jats:italic> has a Hamilton cycle, and therefore a perfect matching (if <jats:italic>n</jats:italic> is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have <jats:italic>many</jats:italic> Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph <jats:italic>G</jats:italic> (in terms of an entropy-like parameter of <jats:italic>G</jats:italic> ). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d<k$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo><</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , and for <jats:italic>n</jats:italic> divisible by <jats:italic>k</jats:italic> , let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the minimum <jats:italic>d</jats:italic> -degree that ensures the existence of a perfect matching in an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph <jats:italic>G</jats:italic> has minimum <jats:italic>d</jats:italic> -degree at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(1+gamma )m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(
极值图论的一个基本定理是狄拉克定理，它说如果一个n顶点图G的最小度至少为n /2，那么G有一个汉密尔顿环，因此是一个完美匹配（如果n是偶数）。后来Sárközy， Selkow和szemer的工作表明，事实上狄拉克图有许多汉密尔顿环和完美匹配，最终Cuckler和Kahn的结果给出了狄拉克图G中汉密尔顿环和完美匹配的数量的精确描述（根据G的类熵参数）。本文将Cuckler和Kahn的结果推广到超图中的完美匹配。对于正整数$$d<k$$ d ＆lt; k，对于n能被k整除，设$$m_{d}(k,n)$$ m d （k, n）为保证n顶点k一致超图中存在完美匹配的最小d度。一般来说，确定（甚至是渐近地）$$m_{d}(k,n)$$ m d （k, n）的值是一个悬而未决的问题，但我们仍然能够证明Cuckler-Kahn定理的一个类比，表明如果一个n顶点k -一致超图G具有最小d度至少$$(1+gamma )m_{d}(k,n)$$ (1 + γ) m d (k, n)（对于任何常数$$gamma >0$$ γ ＆gt; 0），那么G中的完美匹配的数量是由G的一个类似熵的参数控制的。这加强了kang - kelly - k - hn - osthus - pfenninger和Pham-Sah-Sawhney-Simkin的粗略估计。
{"title":"Counting Perfect Matchings in Dirac Hypergraphs","authors":"Matthew Kwan, Roodabeh Safavi, Yiting Wang","doi":"10.1007/s00493-025-00194-8","DOIUrl":"https://doi.org/10.1007/s00493-025-00194-8","url":null,"abstract":"One of the foundational theorems of extremal graph theory is <jats:italic>Dirac’s theorem</jats:italic> , which says that if an <jats:italic>n</jats:italic> -vertex graph <jats:italic>G</jats:italic> has minimum degree at least <jats:italic>n</jats:italic> /2, then <jats:italic>G</jats:italic> has a Hamilton cycle, and therefore a perfect matching (if <jats:italic>n</jats:italic> is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have <jats:italic>many</jats:italic> Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph <jats:italic>G</jats:italic> (in terms of an entropy-like parameter of <jats:italic>G</jats:italic> ). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d<k$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo><</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , and for <jats:italic>n</jats:italic> divisible by <jats:italic>k</jats:italic> , let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the minimum <jats:italic>d</jats:italic> -degree that ensures the existence of a perfect matching in an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph <jats:italic>G</jats:italic> has minimum <jats:italic>d</jats:italic> -degree at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(1+gamma )m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"117 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070700","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}
Pub Date : 2026-01-28DOI: 10.1007/s00493-026-00200-7
Thomas F. Bloom, Jakob Führer, Oliver Roche-Newton
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set A containing $$Omega (|A|^{3/2})$$Ω(|A|3/2) three-term arithmetic progressions.
本文考虑了凸集可加性结构的几种不同测度。主要结果给出了一个包含$$Omega (|A|^{3/2})$$ Ω （| a | 3 / 2）三项等差数列的凸集a的构造。
{"title":"Additive Structure in Convex Sets","authors":"Thomas F. Bloom, Jakob Führer, Oliver Roche-Newton","doi":"10.1007/s00493-026-00200-7","DOIUrl":"https://doi.org/10.1007/s00493-026-00200-7","url":null,"abstract":"This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set <jats:italic>A</jats:italic> containing <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (|A|^{3/2})$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> three-term arithmetic progressions.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"388 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070554","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}
Pub Date : 2026-01-28DOI: 10.1007/s00493-026-00198-y
Jinha Kim
{"title":"Topology of Independence Complexes and Cycle Structure of Hypergraphs","authors":"Jinha Kim","doi":"10.1007/s00493-026-00198-y","DOIUrl":"https://doi.org/10.1007/s00493-026-00198-y","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"238 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070555","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}
Pub Date : 2026-01-28DOI: 10.1007/s00493-026-00199-x
Kristóf Bérczi, Márton Borbényi, László Lovász, László Márton Tóth
We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.
{"title":"Quotient-Convergence of Submodular Setfunctions","authors":"Kristóf Bérczi, Márton Borbényi, László Lovász, László Márton Tóth","doi":"10.1007/s00493-026-00199-x","DOIUrl":"https://doi.org/10.1007/s00493-026-00199-x","url":null,"abstract":"We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"179 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070553","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}
Pub Date : 2026-01-07DOI: 10.1007/s00493-025-00197-5
Elías Mochán
polytopes generalize the face lattice of convex polytopes. A polytope is semiregular if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how these same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks.
{"title":"Semiregular Abstract Polyhedra with Trivial Facet Stabilizer","authors":"Elías Mochán","doi":"10.1007/s00493-025-00197-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00197-5","url":null,"abstract":"<jats:italic> polytopes</jats:italic> generalize the face lattice of convex polytopes. A polytope is <jats:italic>semiregular</jats:italic> if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how these same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"85 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947443","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}
Pub Date : 2026-01-07DOI: 10.1007/s00493-025-00193-9
Chi Hoi Yip
Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted k -th powers. They conjectured that for each $$kge 2$$k≥2 , if one changes $$o(X^{1/k})$$o(X1/k) elements of $$M_k'={x^k+1: x in mathbb {N}}$$Mk′={xk+1:x∈N} up to X , then the resulting set cannot be written as a product set AB nontrivially. In this paper, we confirm a more general version of their conjecture for $$kge 3$$k≥3 .
最近，Hajdu和Sárközy在Erdős关于平方集合的小扰动的加性不可约性的猜想的启发下，研究了平移k次幂的猜想的乘法模拟。他们推测，对于每个$$kge 2$$ k≥2，如果改变$$M_k'={x^k+1: x in mathbb {N}}$$ M k ' = X k + 1的$$o(X^{1/k})$$ o （x1 / k）个元素：X∈N{到X，则结果集不能非平凡地写成乘积集AB。在本文中，我们证实了他们的猜想}$$kge 3$$ k≥3的一个更一般的版本。
{"title":"Multiplicative irreducibility of small perturbations of the set of shifted k-th powers","authors":"Chi Hoi Yip","doi":"10.1007/s00493-025-00193-9","DOIUrl":"https://doi.org/10.1007/s00493-025-00193-9","url":null,"abstract":"Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted <jats:italic>k</jats:italic> -th powers. They conjectured that for each <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$kge 2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , if one changes <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$o(X^{1/k})$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> elements of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$M_k'={x^k+1: x in mathbb {N}}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>:</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> up to <jats:italic>X</jats:italic> , then the resulting set cannot be written as a product set <jats:italic>AB</jats:italic> nontrivially. In this paper, we confirm a more general version of their conjecture for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$kge 3$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> .","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947395","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}
Pub Date : 2026-01-07DOI: 10.1007/s00493-025-00195-7
Liran Rotem, Alon Schejter, Boaz A. Slomka
We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in $$mathbb {C}^n$$Cn , as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in $$mathbb {C}^n$$Cn is equal to $$2^{n+1}-1$$2n+1-1 and that the fractional illumination number of the polydisc in $$mathbb {C}^n$$Cn is equal to $$2^n$$2n . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.
我们在$$mathbb {C}^n$$ C n中为复杂凸体制定了著名的Levi-Hadwiger-Boltyanski照明（或覆盖）猜想的复杂模拟，以及它的（不可比较的）分数版本。提出这些问题的一个关键因素是计算超立方体的复杂模拟（即多面体）的经典光照数和分数光照数。证明了$$mathbb {C}^n$$ C n中多盘的照明数等于$$2^{n+1}-1$$ 2n + 1 - 1, $$mathbb {C}^n$$ C n中多盘的分数照明数等于$$2^n$$ 2n。此外，我们还验证了这两个猜想对于复杂分区和分区类。
{"title":"The complex Illumination problem","authors":"Liran Rotem, Alon Schejter, Boaz A. Slomka","doi":"10.1007/s00493-025-00195-7","DOIUrl":"https://doi.org/10.1007/s00493-025-00195-7","url":null,"abstract":"We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> , as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is equal to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2^{n+1}-1$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and that the fractional illumination number of the polydisc in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is equal to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947444","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}
Pub Date : 2025-12-12DOI: 10.1007/s00493-025-00192-w
Karim Alexander Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson
{"title":"p-Anisotropy on the Moment Curve for Homology Manifolds and Cycles","authors":"Karim Alexander Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson","doi":"10.1007/s00493-025-00192-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00192-w","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"166 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753139","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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