Combinatorica最新文献

{"title":"Exact Results on Traces of Sets","authors":"Mingze Li, Jie Ma, Mingyuan Rong","doi":"10.1007/s00493-026-00209-y","DOIUrl":"https://doi.org/10.1007/s00493-026-00209-y","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147751496","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":"Explosive appearance of cores and bootstrap percolation on lattices","authors":"Ivailo Hartarsky, Lyuben Lichev","doi":"10.1007/s00493-026-00213-2","DOIUrl":"https://doi.org/10.1007/s00493-026-00213-2","url":null,"abstract":"Consider the process where the <jats:italic>n</jats:italic> vertices of a square 2-dimensional torus appear consecutively in a random order. We show that typically the size of the 3-core of the corresponding induced unit-distance graph transitions from 0 to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$n-o(n)$$</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:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from <jats:italic>o</jats:italic> ( <jats:italic>n</jats:italic> ) to <jats:italic>n</jats:italic> . This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman–Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the 2-dimensional square lattice, including neighbourhoods given by large <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ell ^p$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> balls for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$pin [1,infty ]$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> .","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"151 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147751500","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":"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}

{"title":"Modified Macdonald Polynomials and $$mu $$-Mahonian Statistics","authors":"Emma Yu Jin, Xiaowei Lin","doi":"10.1007/s00493-026-00204-3","DOIUrl":"https://doi.org/10.1007/s00493-026-00204-3","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"51 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147461905","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":"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 &lt;jats:italic&gt;n&lt;/jats:italic&gt; -vertex graph is &lt;jats:italic&gt;degree 3-critical&lt;/jats:italic&gt; if it has &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$2n - 2$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; 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: &lt;jats:list list-type=\"bullet\"&gt; &lt;jats:list-item&gt; every &lt;jats:italic&gt;n&lt;/jats:italic&gt; -vertex degree 3-critical graph has &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$Omega (log n)$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;Ω&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; distinct cycle lengths; &lt;/jats:list-item&gt; &lt;jats:list-item&gt; every tree with maximum degree &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$Delta ge 3$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;Δ&lt;/mml:mi&gt; &lt;mml:mo&gt;≥&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$ell $$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;ℓ&lt;/mml:mi&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; leaves has at least &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$log _{Delta -1}, ((Delta -2)ell )$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;Δ&lt;/mml:mi&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mspace/&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;Δ&lt;/mml:mi&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mi&gt;ℓ&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; distinct leaf-to-leaf path lengths; &lt;/jats:list-item&gt; &lt;jats:list-item&gt; for every integer &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$Nge 1$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;N&lt;/mml:mi&gt; &lt;mml:mo&gt;≥&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/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}

{"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}

{"title":"Erdős-Pósa property of A-paths in unoriented group-labelled graphs","authors":"O-joung Kwon, Youngho Yoo","doi":"10.1007/s00493-026-00203-4","DOIUrl":"https://doi.org/10.1007/s00493-026-00203-4","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"45 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147359911","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":"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 &lt;jats:italic&gt;t&lt;/jats:italic&gt; -shatters a set if it induces at least &lt;jats:italic&gt;t&lt;/jats:italic&gt; distinct permutations on that set. What is the minimum number &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$f_k(n,t)$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;f&lt;/mml:mi&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of permutations of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$${1, dots , n}$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;{&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mo&gt;⋯&lt;/mml:mo&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;}&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; that &lt;jats:italic&gt;t&lt;/jats:italic&gt; -shatter all subsets of size &lt;jats:italic&gt;k&lt;/jats:italic&gt; ? For &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$t le 2$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;≤&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$f_k(n,t) = Theta (1)$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;f&lt;/mml:mi&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mi&gt;Θ&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; . Spencer showed that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$f_k(n,t) = Theta (log log n)$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;f&lt;/mml:mi&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mi&gt;Θ&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$3 le t le k$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo&gt;≤&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;≤&lt;/mml:mo&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$f_k(n,k!) = Theta (log n)$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;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}

{"title":"The Classification of Homomorphism Homogeneous Oriented Graphs","authors":"Bojana Pavlica, Christian Pech, Maja Pech","doi":"10.1007/s00493-026-00206-1","DOIUrl":"https://doi.org/10.1007/s00493-026-00206-1","url":null,"abstract":"The modern theory of homogeneous structures begins with the work of Roland Fraïssé. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Nešetřil. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. Among these we identify those that are polymorphism homogeneous. Here an oriented graph is called <italic>polymorphism homogeneous</italic> if each of its finite powers is homomorphism homogeneous.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"20 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147507834","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":"Rigidity Matroids and Linear Algebraic Matroids with Applications to Matrix Completion and Tensor Codes","authors":"Joshua Brakensiek, Manik Dhar, Jiyang Gao, Sivakanth Gopi, Matt Larson","doi":"10.1007/s00493-026-00208-z","DOIUrl":"https://doi.org/10.1007/s00493-026-00208-z","url":null,"abstract":"We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(m, n, a=2, b=2)$$end{document}</tex-math></alternatives></inline-formula> maximally recoverable tensor code.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"14 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147507889","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}