{"title":"Bounded unique representation bases for the integers","authors":"Yong-Gao Chen, Jin-Hui Fang","doi":"10.1016/j.ejc.2024.104080","DOIUrl":"10.1016/j.ejc.2024.104080","url":null,"abstract":"<div><div>For a nonempty set <span><math><mi>A</mi></math></span> of integers and an integer <span><math><mi>n</mi></math></span>, let <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>≤</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>, and let <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>. Erdős and Turán (1941) posed the profound conjecture: if <span><math><mi>A</mi></math></span> is a set of positive integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span> for all sufficiently large <span><math><mi>n</mi></math></span>, then <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set <span><math><mi>A</mi></math></span> with logarithmic growth such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span>. In this paper, we prove that, for any positive function <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn></mrow></math></span> as <span><math><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, there is a bounded set <span><math><mi>A</mi></math></span> of integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all positi","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104080"},"PeriodicalIF":1.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour
{"title":"Induced subgraph density. II. Sparse and dense sets in cographs","authors":"Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.ejc.2024.104075","DOIUrl":"10.1016/j.ejc.2024.104075","url":null,"abstract":"<div><div>A well-known theorem of Rödl says that for every graph <span><math><mi>H</mi></math></span>, and every <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, there exists <span><math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math></span> such that if <span><math><mi>G</mi></math></span> does not contain an induced copy of <span><math><mi>H</mi></math></span>, then there exists <span><math><mrow><mi>X</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>δ</mi><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span> such that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has edge-density at most <span><math><mi>ɛ</mi></math></span>. But how does <span><math><mi>δ</mi></math></span> depend on <span><math><mi>ϵ</mi></math></span>? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>, Rödl’s theorem holds with <span><math><mrow><mi>δ</mi><mo>=</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow></math></span>. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs <span><math><mi>H</mi></math></span>. Our first result shows that it is true when <span><math><mrow><mi>H</mi><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math></span>. Indeed, in that case we can take <span><math><mrow><mi>δ</mi><mo>=</mo><mi>ɛ</mi></mrow></math></span>, and insist that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has maximum degree at most <span><math><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span>).</div><div>Second, we will show that every graph <span><math><mi>H</mi></math></span> that can be obtained by substitution from copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say <span><math><mi>H</mi></math></span> is <em>viral</em> if there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104075"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano
{"title":"The diameter of randomly twisted hypercubes","authors":"Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano","doi":"10.1016/j.ejc.2024.104078","DOIUrl":"10.1016/j.ejc.2024.104078","url":null,"abstract":"<div><div>The <span><math><mi>n</mi></math></span>-dimensional random twisted hypercube <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed recursively by taking two instances of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>/</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability and at least <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span>. We improve their upper bound by showing that <span><math><mrow><mi>diam</mi><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></mfrac></mrow></math></span> with high probability.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104078"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ademir Hujdurović , István Kovács , Klavdija Kutnar , Dragan Marušič
{"title":"Intersection density of transitive groups with small cyclic point stabilizers","authors":"Ademir Hujdurović , István Kovács , Klavdija Kutnar , Dragan Marušič","doi":"10.1016/j.ejc.2024.104079","DOIUrl":"10.1016/j.ejc.2024.104079","url":null,"abstract":"<div><div>For a permutation group <span><math><mi>G</mi></math></span> acting on a set <span><math><mi>V</mi></math></span>, a subset <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span> is said to be an <em>intersecting set</em> if for every pair of elements <span><math><mrow><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>F</mi></mrow></math></span> there exists <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> such that <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. The <em>intersection density</em> <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a transitive permutation group <span><math><mi>G</mi></math></span> is the maximum value of the quotient <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>/</mo><mrow><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span> where <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> is a stabilizer of a point <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mi>F</mi></math></span> runs over all intersecting sets in <span><math><mi>G</mi></math></span>. If <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> is a largest intersecting set in <span><math><mi>G</mi></math></span> then <span><math><mi>G</mi></math></span> is said to have the <em>Erdős-Ko-Rado (EKR)-property</em>. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among others, constructions of an infinite family of transitive permutation groups having point stabilizer of order 3 with intersection density <span><math><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></math></span> and of infinite families of transitive permutation groups having point stabilizer of order 3 with arbitrarily large intersection density are given.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104079"},"PeriodicalIF":1.0,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Turán numbers of ordered tight hyperpaths","authors":"John P. Bright, Kevin G. Milans, Jackson Porter","doi":"10.1016/j.ejc.2024.104070","DOIUrl":"10.1016/j.ejc.2024.104070","url":null,"abstract":"<div><div>An <em>ordered hypergraph</em> is a hypergraph <span><math><mi>G</mi></math></span> whose vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is linearly ordered. We find the Turán numbers for the <span><math><mi>r</mi></math></span>-uniform <span><math><mi>s</mi></math></span>-vertex tight path <span><math><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup></math></span> (with vertices in the natural order) exactly when <span><math><mrow><mi>r</mi><mo>≤</mo><mi>s</mi><mo><</mo><mn>2</mn><mi>r</mi></mrow></math></span> and <span><math><mi>n</mi></math></span> is even; our results imply <span><math><mrow><mover><mrow><mi>ex</mi></mrow><mo>→</mo></mover><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi><mo>−</mo><mi>r</mi></mrow></msup></mrow></mfrac><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> when <span><math><mrow><mi>r</mi><mo>≤</mo><mi>s</mi><mo><</mo><mn>2</mn><mi>r</mi></mrow></math></span>. When <span><math><mrow><mi>s</mi><mo>≥</mo><mn>2</mn><mi>r</mi></mrow></math></span>, the asymptotics of <span><math><mrow><mover><mrow><mi>ex</mi></mrow><mo>→</mo></mover><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow></mrow></math></span> remain open. For <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span>, we give a construction of an <span><math><mi>r</mi></math></span>-uniform <span><math><mi>n</mi></math></span>-vertex hypergraph not containing <span><math><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup></math></span> which we conjecture to be asymptotically extremal.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104070"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
John Haslegrave , Alex Scott , Youri Tamitegama , Jane Tan
{"title":"Boundary rigidity of 3D CAT(0) cube complexes","authors":"John Haslegrave , Alex Scott , Youri Tamitegama , Jane Tan","doi":"10.1016/j.ejc.2024.104077","DOIUrl":"10.1016/j.ejc.2024.104077","url":null,"abstract":"<div><div>The boundary rigidity problem is a classical question from Riemannian geometry: if <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></math></span> is a Riemannian manifold with smooth boundary, is the geometry of <span><math><mi>M</mi></math></span> determined up to isometry by the metric <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> induced on the boundary <span><math><mrow><mi>∂</mi><mi>M</mi></mrow></math></span>? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104077"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke
{"title":"Decks of rooted binary trees","authors":"Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke","doi":"10.1016/j.ejc.2024.104076","DOIUrl":"10.1016/j.ejc.2024.104076","url":null,"abstract":"<div><div>We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree <span><math><mi>T</mi></math></span> refers to the set (resp. multiset) of leaf-induced binary subtrees of <span><math><mi>T</mi></math></span>. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on <span><math><mi>n</mi></math></span> leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for <span><math><mi>k</mi></math></span>-universal trees, i.e., rooted binary trees that contain all <span><math><mi>k</mi></math></span>-leaf rooted binary trees as leaf-induced subtrees.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104076"},"PeriodicalIF":1.0,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole","authors":"Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl","doi":"10.1016/j.ejc.2024.104074","DOIUrl":"10.1016/j.ejc.2024.104074","url":null,"abstract":"<div><div>A <em>clock</em> is a graph consisting of an induced cycle <span><math><mi>C</mi></math></span> and a vertex not in <span><math><mi>C</mi></math></span> with at least two non-adjacent neighbours in <span><math><mi>C</mi></math></span>. We show that every clock-free graph of large treewidth contains a “basic obstruction” of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104074"},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001598/pdfft?md5=fa98c8a13265d848775c4b52beb995da&pid=1-s2.0-S0195669824001598-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Zig-zag Eulerian polynomials","authors":"T. Kyle Petersen , Yan Zhuang","doi":"10.1016/j.ejc.2024.104073","DOIUrl":"10.1016/j.ejc.2024.104073","url":null,"abstract":"<div><div>For any finite partially ordered set <span><math><mi>P</mi></math></span>, the <span><math><mi>P</mi></math></span>-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of <span><math><mi>P</mi></math></span>, and is closely related to the order polynomial of <span><math><mi>P</mi></math></span> arising in the theory of <span><math><mi>P</mi></math></span>-partitions. Here we study the <span><math><mi>P</mi></math></span>-Eulerian polynomial where <span><math><mi>P</mi></math></span> is a naturally labeled zig-zag poset; we call these <em>zig-zag Eulerian polynomials</em>. A result of Brändén implies that these polynomials are gamma-nonnegative, and hence their coefficients are symmetric and unimodal. The zig-zag Eulerian polynomials and the associated order polynomials have appeared fleetingly in the literature in a wide variety of contexts—e.g., in the study of polytopes, magic labelings of graphs, and Kekulé structures—but they do not appear to have been studied systematically.</div><div>In this paper, we use a “relaxed” version of <span><math><mi>P</mi></math></span>-partitions to both survey and unify results. Our technique shows that the zig-zag Eulerian polynomials also capture the distribution of “big returns” over the set of (up-down) alternating permutations, as first observed by Coons and Sullivant. We develop recurrences for refined versions of the relevant generating functions, which evoke similarities to recurrences for the classical Eulerian polynomials. We conclude with a literature survey and open questions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104073"},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001586/pdfft?md5=899109a09f3bf833016e01aae31a7980&pid=1-s2.0-S0195669824001586-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On non-degenerate Turán problems for expansions","authors":"Dániel Gerbner","doi":"10.1016/j.ejc.2024.104071","DOIUrl":"10.1016/j.ejc.2024.104071","url":null,"abstract":"<div><div>The <span><math><mi>r</mi></math></span>-uniform expansion <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span> of a graph <span><math><mi>F</mi></math></span> is obtained by enlarging each edge with <span><math><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></math></span> new vertices such that altogether we use <span><math><mrow><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> new vertices. Two simple lower bounds on the largest number <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>r</mi></math></span>-edges in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span>-free <span><math><mi>r</mi></math></span>-graphs are <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> (in the case <span><math><mi>F</mi></math></span> is not a star) and <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which is the largest number of <span><math><mi>r</mi></math></span>-cliques in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs. We prove that <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The proof comes with a structure theorem that we use to determine <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> exactly for some graphs <span><math><mi>F</mi></math></span>, every <span><math><mrow><mi>r</mi><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104071"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001562/pdfft?md5=86fa8d5991cc3c3ff302bc8fdbd50279&pid=1-s2.0-S0195669824001562-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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