European Journal of Combinatorics最新文献

{"title":"On two conjectures of Shallit about Thue–Morse-like sequences","authors":"Lubomíra Dvořáková ,&nbsp;Savinien Kreczman ,&nbsp;Edita Pelantová","doi":"10.1016/j.ejc.2025.104250","DOIUrl":"10.1016/j.ejc.2025.104250","url":null,"abstract":"<div><div>We study a class of infinite words <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, recently introduced by J. Shallit. This class includes the Thue–Morse sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, the Fibonacci–Thue–Morse sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and the Allouche–Johnson sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Shallit stated and for <span><math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math></span> proved two conjectures on properties of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every <span><math><mi>k</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104250"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247905","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":"On minimal actions of countable groups","authors":"Eli Glasner ,&nbsp;Benjamin Weiss","doi":"10.1016/j.ejc.2025.104261","DOIUrl":"10.1016/j.ejc.2025.104261","url":null,"abstract":"<div><div>Our purpose here is to review some recent developments in the theory of dynamical systems whose common theme is a link between minimal dynamical systems, certain Ramsey type combinatorial properties, and the Lovász local lemma (LLL). For a general countable group <span><math><mi>G</mi></math></span> the two classes of minimal systems we will deal with are (I) the minimal subsystems of the <em>subgroup system</em> <span><math><mrow><mo>(</mo><mi>Sub</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>G</mi><mo>)</mo></mrow></math></span>, called URS’s (uniformly recurrent subgroups), and (II) minimal <em>subshifts</em>; i.e. subsystems of the binary Bernoulli <span><math><mi>G</mi></math></span>-shift <span><math><mrow><mo>(</mo><msup><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow><mrow><mi>G</mi></mrow></msup><mo>,</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></msub><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104261"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623514","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":"Characterizing avoidance in cycles via vincular patterns","authors":"Robert P. Laudone","doi":"10.1016/j.ejc.2025.104249","DOIUrl":"10.1016/j.ejc.2025.104249","url":null,"abstract":"<div><div>We show that cyclic permutations avoiding 321 are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of 321 avoidance in terms of equality of the upper bound of the Daiconis–Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of 321 avoiding cycles.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104249"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326522","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":"On the structures of subset sums in higher dimension","authors":"Norbert Hegyvári ,&nbsp;Máté Pálfy ,&nbsp;Erfei Yue","doi":"10.1016/j.ejc.2025.104279","DOIUrl":"10.1016/j.ejc.2025.104279","url":null,"abstract":"<div><div>A given subset <span><math><mi>A</mi></math></span> of natural numbers is said to be complete if every sufficiently large integer of <span><math><mi>N</mi></math></span> is the sum of distinct terms taken from <span><math><mi>A</mi></math></span>. In higher dimension the definition is similar: for any <span><math><mrow><mi>X</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>}</mo></mrow><mo>⊆</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> let <span><span><span><math><mrow><mi>F</mi><mi>S</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>{</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mspace></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>X</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo><mspace></mspace><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>&lt;</mo><mi>∞</mi><mo>}</mo></mrow><mo>.</mo></mrow></math></span></span></span>We say that a set <span><math><mi>X</mi></math></span> is <em>complete respect to the region</em> <span><math><mrow><mi>R</mi><mo>⊆</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> if <span><math><mrow><mi>R</mi><mo>⊆</mo><mi>F</mi><mi>S</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> holds. A set <span><math><mi>X</mi></math></span> is a <em>thin complete set</em> of <span><math><mi>R</mi></math></span> if the counting function <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>≤</mo><mi>k</mi><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>R</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></math></span> for some <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and <span><math><mrow><mi>F</mi><mi>S</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>⊇</mo><mi>R</mi></mrow></math></span>. We construct ‘thin’ complete set provided the domain <span><math><mi>R</mi></math></span> does not contain half-lines parallel to the axis. Furthermore we investigate the distribution of the subset sum of ‘splitable’ sets too.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104279"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467313","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":"Flips in two-dimensional hypertriangulations","authors":"Herbert Edelsbrunner ,&nbsp;Alexey Garber ,&nbsp;Mohadese Ghafari ,&nbsp;Teresa Heiss ,&nbsp;Morteza Saghafian","doi":"10.1016/j.ejc.2025.104248","DOIUrl":"10.1016/j.ejc.2025.104248","url":null,"abstract":"<div><div>We study flips in hypertriangulations of planar points sets. Here a level-<span><math><mi>k</mi></math></span> hypertriangulation of <span><math><mi>n</mi></math></span> points in the plane is a subdivision induced by the projection of a <span><math><mi>k</mi></math></span>-hypersimplex, which is the convex hull of the barycenters of the <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional faces of the standard <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104248"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247903","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":"Ramsey expansions of metrically homogeneous graphs","authors":"Andrés Aranda ,&nbsp;David Bradley-Williams ,&nbsp;Jan Hubička ,&nbsp;Miltiadis Karamanlis ,&nbsp;Michael Kompatscher ,&nbsp;Matěj Konečný ,&nbsp;Micheal Pawliuk","doi":"10.1016/j.ejc.2025.104255","DOIUrl":"10.1016/j.ejc.2025.104255","url":null,"abstract":"<div><div>We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin’s catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA.</div><div>Our results are a contribution to Nešetřil’s classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin’s classes. The existence of such a “completion algorithm” then allows us to apply several strong results in the areas that imply EPPA or the Ramsey property.</div><div>The main results have numerous consequences for the automorphism groups of the Fraïssé limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre’s property (FA).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104255"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623508","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":"Ramsey expansions of Λ-ultrametric spaces","authors":"Samuel Braunfeld","doi":"10.1016/j.ejc.2025.104257","DOIUrl":"10.1016/j.ejc.2025.104257","url":null,"abstract":"<div><div>For a finite lattice <span><math><mi>Λ</mi></math></span>, <span><math><mi>Λ</mi></math></span>-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When <span><math><mi>Λ</mi></math></span> is finite and distributive, there exists a generic <span><math><mi>Λ</mi></math></span>-ultrametric space, and we here identify a family of Ramsey expansions for that space. This then allows a description the universal minimal flow of its automorphism group, and also implies the Ramsey property for all homogeneous finite-dimensional permutation structures, i.e., homogeneous structures in a language of finitely many linear orders. A point of technical interest is that our proof involves classes with non-unary algebraic closure operations. As a byproduct of some of the concepts developed, we also arrive at a natural description of all homogeneous finite-dimensional permutation structures.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104257"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623510","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":"Forbidden cycles in metrically homogeneous graphs","authors":"Jan Hubička ,&nbsp;Michael Kompatscher ,&nbsp;Matěj Konečný","doi":"10.1016/j.ejc.2025.104262","DOIUrl":"10.1016/j.ejc.2025.104262","url":null,"abstract":"<div><div>In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space <span><math><mi>Γ</mi></math></span> of finite diameter <span><math><mi>δ</mi></math></span> from Cherlin’s catalogue of metrically homogeneous graphs, there exists a finite family <span><math><mi>F</mi></math></span> of <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled cycles such that a <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled graph is a subgraph of <span><math><mi>Γ</mi></math></span> if and only if it contains no homomorphic images of cycles from <span><math><mi>F</mi></math></span>. However, the cycles in the families <span><math><mi>F</mi></math></span> were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families <span><math><mi>F</mi></math></span>, heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenisations of <span><math><mi>ω</mi></math></span>-categorical <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104262"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623515","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":"Finite Ramsey degrees and Fraïssé expansions with the Ramsey property","authors":"Lionel Nguyen Van Thé","doi":"10.1016/j.ejc.2025.104264","DOIUrl":"10.1016/j.ejc.2025.104264","url":null,"abstract":"<div><div>By a result of Zucker, every Fraïssé structure <span><math><mi>F</mi></math></span> for which the elements of <span><math><mrow><mi>Age</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> have finite Ramsey degrees admits a Fraïssé precompact expansion <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> whose age <span><math><mrow><mi>Age</mi><mrow><mo>(</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> has the Ramsey property. While the original method uses dynamics in spaces of ultrafilters, the purpose of the present short note is to provide a different proof, based on classical tools from Fraïssé theory.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104264"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624161","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":"Higher q-continued fractions","authors":"Amanda Burcroff ,&nbsp;Nicholas Ovenhouse ,&nbsp;Ralf Schiffler ,&nbsp;Sylvester W. Zhang","doi":"10.1016/j.ejc.2025.104244","DOIUrl":"10.1016/j.ejc.2025.104244","url":null,"abstract":"<div><div>We introduce a <span><math><mi>q</mi></math></span>-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the <span><math><mi>q</mi></math></span>-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for <span><math><mi>P</mi></math></span>-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the <span><math><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></math></span> case. We also show that certain properties enjoyed by the <span><math><mi>q</mi></math></span>-rationals are also satisfied by our higher versions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104244"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096502","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}