Pub Date : 2026-03-12DOI: 10.1007/s00493-026-00202-5
Anurag Bishnoi, István Tomon
A strong s -blocking set in a projective space is a set of points that intersects each codimension- s subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $$(k - 1)$$(k-1) -dimensional projective space over $$mathbb {F}_q$$Fq of size $$O_s(q^s k)$$Os(qsk) , which is optimal up to the constant factor depending on s . This also yields an optimal explicit construction of affine blocking sets in $$mathbb {F}_q^k$$Fqk with respect to codimension- $$(s+1)$$(s+1) affine subspaces, and of s -minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong 1-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong s -blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.
射影空间中的强s块集是与子空间中的每个余维- s子空间相交的点的集合。我们在大小为$$O_s(q^s k)$$ O s （q s k）的$$mathbb {F}_q$$ F q上的$$(k - 1)$$ （k - 1）维投影空间中给出了这种集合的显式构造，该构造在依赖于s的常数因子上是最优的。这也产生了$$mathbb {F}_q^k$$ F q k中关于余维- $$(s+1)$$ （s + 1）仿射子空间和s -最小码的仿射块集的最优显式构造。我们的方法是由最近的强1块集的Alon， Bishnoi， Das和Neri的构造所激发的，它使用带有精心选择的向量集作为其顶点集的展开图。我们工作的主要新颖之处在于在这些扩展图之上构造特定的超图，其中树状结构对应于强s块集。我们还讨论了与超图的大小拉姆齐数的一些联系，这可能是独立的兴趣。
{"title":"Explicit Constructions of Optimal Blocking Sets and Minimal Codes","authors":"Anurag Bishnoi, István Tomon","doi":"10.1007/s00493-026-00202-5","DOIUrl":"https://doi.org/10.1007/s00493-026-00202-5","url":null,"abstract":"A strong <jats:italic>s</jats:italic> -blocking set in a projective space is a set of points that intersects each codimension- <jats:italic>s</jats:italic> subspace in a spanning set of the subspace. We present an explicit construction of such sets in a <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(k - 1)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -dimensional projective space over <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {F}_q$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> of size <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O_s(q^s k)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , which is optimal up to the constant factor depending on <jats:italic>s</jats:italic> . This also yields an optimal explicit construction of affine blocking sets in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {F}_q^k$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> </mml:math> </jats:alternatives> </jats:inline-formula> with respect to codimension- <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(s+1)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> affine subspaces, and of <jats:italic>s</jats:italic> -minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong 1-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong <jats:italic>s</jats:italic> -blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"266 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147461906","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 : 2026-03-04DOI: 10.1007/s00493-026-00205-2
Francesco Di Braccio, Kyriakos Katsamaktsis, Jie Ma, Alexandru Malekshahian, Ziyuan Zhao
An <jats:italic>n</jats:italic> -vertex graph is <jats:italic>degree 3-critical</jats:italic> if it has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2n - 2$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: <jats:list list-type="bullet"> <jats:list-item> every <jats:italic>n</jats:italic> -vertex degree 3-critical graph has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (log n)$$</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>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct cycle lengths; </jats:list-item> <jats:list-item> every tree with maximum degree <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Delta ge 3$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ell $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> leaves has at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$log _{Delta -1}, ((Delta -2)ell )$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>log</mml:mo> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct leaf-to-leaf path lengths; </jats:list-item> <jats:list-item> for every integer <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Nge 1$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternative
{"title":"Leaf-to-leaf paths and cycles in degree-critical graphs","authors":"Francesco Di Braccio, Kyriakos Katsamaktsis, Jie Ma, Alexandru Malekshahian, Ziyuan Zhao","doi":"10.1007/s00493-026-00205-2","DOIUrl":"https://doi.org/10.1007/s00493-026-00205-2","url":null,"abstract":"An <jats:italic>n</jats:italic> -vertex graph is <jats:italic>degree 3-critical</jats:italic> if it has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2n - 2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: <jats:list list-type=\"bullet\"> <jats:list-item> every <jats:italic>n</jats:italic> -vertex degree 3-critical graph has <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (log n)$$</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>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct cycle lengths; </jats:list-item> <jats:list-item> every tree with maximum degree <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Delta ge 3$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ell $$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> leaves has at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$log _{Delta -1}, ((Delta -2)ell )$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mo>log</mml:mo> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> distinct leaf-to-leaf path lengths; </jats:list-item> <jats:list-item> for every integer <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Nge 1$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternative","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"55 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147359909","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 : 2026-03-04DOI: 10.1007/s00493-026-00207-0
Hidde Koerts, Benjamin Moore, Sophie Spirkl
{"title":"Orientations of Cycles in Digraphs of High Chromatic Number and High Minimum Out-Degree","authors":"Hidde Koerts, Benjamin Moore, Sophie Spirkl","doi":"10.1007/s00493-026-00207-0","DOIUrl":"https://doi.org/10.1007/s00493-026-00207-0","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"102 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147359910","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 : 2026-03-04DOI: 10.1007/s00493-026-00201-6
António Girão, Lukas Michel, Youri Tamitegama
We say that a family of permutations <jats:italic>t</jats:italic> -shatters a set if it induces at least <jats:italic>t</jats:italic> distinct permutations on that set. What is the minimum number <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of permutations of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${1, dots , n}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> that <jats:italic>t</jats:italic> -shatter all subsets of size <jats:italic>k</jats:italic> ? For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$t le 2$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t) = Theta (1)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . Spencer showed that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t) = Theta (log log n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$3 le t le k$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,k!) = Theta (log n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:ms
我们说，如果一组排列在一个集合上产生至少t个不同的排列，那么它就打破了这个集合。让所有大小为k的子集被t打散的$${1, dots , n}$$ 1,⋯,n 的排列的最小个数$$f_k(n,t)$$ f k （n, t）是多少？当{}$$t le 2$$ t≤2时，$$f_k(n,t) = Theta (1)$$ f k (n, t) = Θ(1)。Spencer证明了$$f_k(n,t) = Theta (log log n)$$ f k (n, t) = Θ （log log n）对于$$3 le t le k$$ 3≤t≤k和$$f_k(n,k!) = Theta (log n)$$ f k （n, k !）= Θ （log n）。1996年，f redi提出了一个问题，即带有排列的部分破碎是否一定总是属于这三种状态之一。Johnson和Wickes最近肯定地解决了$$k = 3$$ k = 3的问题，并证明了$$f_k(n,t) = Theta (log n)$$ f k (n, t) = Θ （log n）对于$$t > 2 (k-1)!$$ t ＆gt; 2 (k - 1) ！． 我们通过证明$$k ge 4$$ k≥4时存在第四态，给出了一个令人惊讶的关于 redi问题的否定答案。我们建立了$$f_k(n,t) = Theta (sqrt{log n})$$ f k (n, t) = Θ （log n）对于t的某些值，并证明了这是$$k = 4$$ k = 4时唯一的另一个区域。我们还证明了$$f_k(n,t) = Theta (log n)$$ f k (n, t) = Θ （log n）对于$$t > 2^{k-1}$$ t ＆gt; 2k - 1。这大大缩小了t的范围，其中$$f_k(n,t)$$ f k （n, t）的渐近行为是未知的。
{"title":"Small Families of Partially Shattering Permutations","authors":"António Girão, Lukas Michel, Youri Tamitegama","doi":"10.1007/s00493-026-00201-6","DOIUrl":"https://doi.org/10.1007/s00493-026-00201-6","url":null,"abstract":"We say that a family of permutations <jats:italic>t</jats:italic> -shatters a set if it induces at least <jats:italic>t</jats:italic> distinct permutations on that set. What is the minimum number <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of permutations of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${1, dots , n}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> that <jats:italic>t</jats:italic> -shatter all subsets of size <jats:italic>k</jats:italic> ? For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$t le 2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t) = Theta (1)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . Spencer showed that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,t) = Theta (log log n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$3 le t le k$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f_k(n,k!) = Theta (log n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:ms","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"69 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147359912","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/s00493-025-00194-8
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
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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}
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