European Journal of Combinatorics最新文献

英文中文

Ramsey expansions of Λ-ultrametric spaces Λ-ultrametric空间的Ramsey展开式

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-16 DOI: 10.1016/j.ejc.2025.104257

Samuel Braunfeld

For a finite lattice

Λ

Λ

-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When

Λ

is finite and distributive, there exists a generic

Λ

-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.

对于有限晶格Λ， Λ-ultrametric空间是描述具有一系列等价关系的结构的方便语言。当Λ是有限且可分配的，存在一个一般的Λ-ultrametric空间，我们在此确定该空间的Ramsey展开式族。这就允许对其自同构群的普遍极小流的描述，同时也暗示了所有齐次有限维排列结构的Ramsey性质，即有限多个线性阶语言中的齐次结构。技术上的一个有趣点是，我们的证明涉及到具有非一元代数闭包操作的类。作为发展的一些概念的副产品，我们也得到了所有齐次有限维排列结构的自然描述。

引用次数: 0

A perfect expansion property 一个完美的展开性质

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-15 DOI: 10.1016/j.ejc.2025.104265

Micheal Pawliuk

We present two exact versions of the quantitative expansion property first presented in Angel et al. (2014), called the Perfect Expansion Property and the disjoint Perfect Expansion Property (PEP and DPEP). This gives a direct combinatorial way of establishing the unique ergodicity of automorphism groups of Fraïssé classes, without having to use the probabilistic arguments in Angel et al. (2014).

We focus on the special case of

D_{n}

, the class of complete,

n

-partite digraphs. Not all structures in this class have the PEP and we classify which structures have the stronger DPEP. The structures with this expansion property are intimately connected with the definable geometric structure of a Fraïssé structure.

We also look at the PEP for semigeneric digraphs, but we do not settle the question of unique ergodicity of the automorphism group of the semigeneric digraph.¹ Surprisingly, there are non-trivial substructures of the semigeneric digraph with the PEP.

我们给出了Angel等人（2014）首次提出的两个精确版本的定量展开性质，称为完美展开性质和不相交完美展开性质（PEP和DPEP）。这提供了一种直接的组合方式来建立Fraïssé类的自同构群的唯一遍历性，而不必使用Angel等人（2014）中的概率参数。我们专注于特殊情况Dn，一类完全的，n部有向图。并不是所有的结构都有PEP，我们对哪些结构有更强的DPEP进行分类。具有这种展开性质的结构与Fraïssé结构的可定义几何结构密切相关。我们也研究了半同型有向图的PEP，但是我们没有解决半同型有向图的自同构群的唯一遍历性问题令人惊讶的是，具有PEP的半同属有向图存在非平凡的子结构。

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引用次数: 0

Reconfiguring homomorphisms to reflexive graphs via a simple reduction 通过简单的约简将同态重新配置为自反图

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-15 DOI: 10.1016/j.ejc.2025.104251

Moritz Mühlenthaler , Mark Siggers , Thomas Suzan

Given a graph

G

and two graph homomorphisms

α

and

β

from

G

to a fixed graph

H

, the problem

H

-recoloring asks whether there is a transformation from

α

β

that changes the image of a single vertex at each step and keeps a graph homomorphism throughout. The complexity of the problem depends, among other things, on the presence of loops on the vertices. We provide a simple reduction that, using a known algorithmic result for

H

-recoloring for square-free irreflexive graphs

H

, yields a polynomial-time algorithm for

H

-recoloring for square-free reflexive graphs

H

. This generalizes all known algorithmic results for

H

-recoloring for reflexive graphs

H

. Furthermore, the construction allows us to reprove some of the known hardness results. Finally, we provide a partial inverse of the construction for bipartite instances.

给定一个图G和从G到固定图H的两个图同态α和β， H-重着色问题是问是否存在从α到β的变换，该变换在每一步改变单个顶点的像并始终保持图同态。这个问题的复杂性取决于顶点上是否存在循环。我们提供了一个简单的约简，使用已知的无平方自反图H的H-重着色算法结果，得到了一个多项式时间算法，用于无平方自反图H的H-重着色。这推广了所有已知的自反图H的H-重着色算法结果。此外，该构造允许我们对一些已知的硬度结果进行修正。最后，我们给出了二部实例的部分逆构造。

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引用次数: 0

Note on a problem of Sárközy on multiplicative representation functions 关于乘法表示函数Sárközy问题的说明

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-15 DOI: 10.1016/j.ejc.2025.104268

Yuchen Ding

Motivated by a 2001 problem of Sárközy, we classify all situations of the integers

b, c, e

and

f

satisfying

\underset{n \to \infty}{lim sup} | d (A, b n + c) - d (A, e n + f) | = \infty

for any infinite

A \subset N

, where

d (A, m) = # {a \in A : a | m} .

受2001年Sárközy问题的启发，我们对任意无限a∧N满足lim supn→∞|d(a,bn+c) - d(a,en+f)|=∞的整数b，c，e和f的所有情况进行分类，其中d(a,m)=#{a∈a:a|m}。

引用次数: 0

Finite Ramsey degrees and Fraïssé expansions with the Ramsey property 有限Ramsey度和Fraïssé Ramsey性质展开式

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-14 DOI: 10.1016/j.ejc.2025.104264

Lionel Nguyen Van Thé

By a result of Zucker, every Fraïssé structure

F

for which the elements of

Age (F)

have finite Ramsey degrees admits a Fraïssé precompact expansion

F^{*}

whose age

Age (F^{*})

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.

由Zucker的结果，每个年龄(F)的元素具有有限Ramsey度的Fraïssé结构F都承认一个年龄Age(F)具有Ramsey性质的Fraïssé预紧展开F *。虽然最初的方法在超滤空间中使用动力学，但目前的简短说明的目的是提供一种基于Fraïssé理论的经典工具的不同证明。

引用次数: 0

A spectral lower bound on chromatic numbers using p-energy 利用p能量的色数的谱下界

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-13 DOI: 10.1016/j.ejc.2025.104252

Clive Elphick , Quanyu Tang , Shengtong Zhang

<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> be the adjacency matrix of a simple graph <span><math><mi>G</mi></math></span>, and let <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For <span><math><mrow><mi>p</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, we define the positive and negative <span><math><mi>p</mi></math></span>-energies of <span><math><mi>G</mi></math></span> by <span><span><span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></munder><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>,</mo><mspace></mspace><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mn>0</mn></mrow></munder><msup><mrow><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> are the eigenvalues of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>. We prove that for all <span><math><mrow><mi>p</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, <span><span><span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfenced><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mfenced><mo>≥</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn><mspace></mspace><mo>+</mo><mo>max</mo><msup

设AG为简单图G的邻接矩阵，χ(G)、χf(G)、χq(G)、ξ(G)、ξf(G)分别表示它的色数、分数色数、量子色数、正交秩和射影秩。对于p≥0，我们定义了G的正负p能：Ep+(G)=∑λi>0λip,Ep−(G)=∑λi<0|λi|p，其中λ1≥⋯≥λn为AG的特征值。我们证明所有p≥0,χ(G)≥χf (G),χq (G),ξ(G)≥ξf (G)≥1 + maxEp + (G) Ep−(G), Ep−(G) Ep + (G) 1 | p−1 |。此结果统一并强化了p∈{0,2，∞}所对应的一系列已有界。特别地，当p=0时，得到惯性界χf(G)≥ξf(G)≥1+maxn+n -,n - n+，其中n+和n -分别表示AG的正特征值和负特征值的个数。这解决了Elphick和Wocjan的两个猜想。我们还证明了对于某些图，p的非整数值提供比现有谱界更清晰的下界。作为一个例子，我们确定了Tilley图的χq，这不能使用现有的（未加权的）p-能界来实现。我们的证明采用了一种新颖的线性代数和测量理论工具的综合，这使我们能够超越现有的谱界。

引用次数: 0

Borel sets of Rado graphs and Ramsey’s theorem Rado图的Borel集与Ramsey定理

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-13 DOI: 10.1016/j.ejc.2025.104260

Natasha Dobrinen

The well-known Galvin-Prikry Theorem (Galvin and Prikry, 1973) states that Borel subsets of the Baire space are Ramsey: Given any Borel subset

X \subseteq {[ω]}^{ω}

, where

{[ω]}^{ω}

is endowed with the metric topology, each infinite subset

X \subseteq ω

contains an infinite subset

Y \subseteq X

such that

{[Y]}^{ω}

is either contained in

X

or disjoint from

X

. Kechris, Pestov, and Todorcevic point out in Kechris et al. (2005) the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from work in Laflamme (2006). The proof uses techniques developed for the author’s work on the Ramsey theory of the Henson graphs (Dobrinen, 2020 and Dobrinen, 2023) as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.

著名的Galvin-Prikry定理（Galvin and Prikry, 1973）指出，Baire空间的Borel子集是Ramsey：给定任意一个Borel子集X≤[ω]ω，其中[ω]ω被赋予度量拓扑，则每一个无限子集X∈一个无限子集Y≤X，使得[Y]ω要么包含在X中，要么与X不相交。Kechris、Pestov和Todorcevic在Kechris et al.（2005）中指出，对于齐次结构缺乏类似的结果。这些结果是寻找具有无限维Ramsey性质的结构与拓扑动力学之间的对应关系的更大目标的必要步骤，扩展了Ramsey性质与极端柔顺性之间的对应关系。在本文中，我们证明了Rado图的Galvin-Prikry定理的一个类比。任何这样的无限维拉姆齐定理都受到Laflamme（2006）工作的约束。该证明使用了作者在Henson图的Ramsey理论（Dobrinen， 2020和Dobrinen， 2023）的工作中开发的技术，以及一些用于融合序列的新方法，用于绕过Baire空间所缺乏的某些合并性质。

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引用次数: 0

On minimal actions of countable groups 论可数群的最小作用

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-13 DOI: 10.1016/j.ejc.2025.104261

Eli Glasner , Benjamin Weiss

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

G

the two classes of minimal systems we will deal with are (I) the minimal subsystems of the subgroup system

(Sub (G), G)

, called URS’s (uniformly recurrent subgroups), and (II) minimal subshifts; i.e. subsystems of the binary Bernoulli

G

-shift

({0, 1}^{G}, {σ_{g}}_{g \in G})

我们的目的是回顾动力系统理论的一些最新进展，这些理论的共同主题是最小动力系统、某些拉姆齐型组合性质和Lovász局部引理（LLL）之间的联系。对于一般可数群G，我们将处理的两类最小系统是(I)子群系统的最小子系统（Sub(G)，G），称为URS（一致循环子群），以及（II）最小子移；即二元伯努利G-shift （{0,1}G,{σg} G∈G）的子系统。

引用次数: 0

First order logic and twin-width in tournaments and dense oriented graphs 竞赛图和密集面向图中的一阶逻辑和双宽度

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-10 DOI: 10.1016/j.ejc.2025.104247

Colin Geniet , Stéphan Thomassé

We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments

T

, first-order model checking is either fixed parameter tractable or

AW [*]

-hard. This dichotomy coincides with the fact that

T

has either bounded or unbounded twin-width, and that the growth of

T

is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions:

T

has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs.

The key for these results is a polynomial time algorithm that takes as input a tournament

T

and computes a linear order

<

V (T)

such that the twin-width of the birelation

(T, <)

is at most some function of the twin-width of

T

. Since approximating twin-width can be done in polynomial time for an ordered structure

(T, <)

, this provides a polynomial time approximation of twin-width for tournaments.

Our results extend to oriented graphs with stable sets of bounded size, which may also be augmented by arbitrary binary relations.

我们用易处理的一阶模型检验来描述比赛的类别。对于每一个世袭类T，一阶模型检验要么是固定参数可处理的，要么是AW[∗]-困难的。这种二分法与以下事实相一致：T要么有界要么无界双宽，T的增长要么最多是指数增长，要么至少是阶乘增长。从模型理论的角度，我们证明了竞赛的NIP类与有界双宽度重合。双宽度也以三个无限的障碍族为特征：当且仅当它从每个族中至少排除一个锦标赛时，T具有有界的双宽度。这推广了Bonnet等人关于有序图的结果。这些结果的关键是多项式时间算法，该算法将锦标赛T作为输入，并计算V(T)上的线性阶数<；，使得双相关（T,<）的双宽度至多是T的双宽度的某个函数。由于对有序结构（T,<）可以在多项式时间内逼近双宽度，因此这为锦标赛提供了双宽度的多项式时间逼近。我们的结果推广到具有稳定的有界大小集的有向图，它也可以被任意的二元关系扩充。

{"title":"First order logic and twin-width in tournaments and dense oriented graphs","authors":"Colin Geniet , Stéphan Thomassé","doi":"10.1016/j.ejc.2025.104247","DOIUrl":"10.1016/j.ejc.2025.104247","url":null,"abstract":"<div><div>We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments <span><math><mi>T</mi></math></span>, first-order model checking is either fixed parameter tractable or <span><math><mrow><mtext>AW</mtext><mrow><mo>[</mo><mo>∗</mo><mo>]</mo></mrow></mrow></math></span>-hard. This dichotomy coincides with the fact that <span><math><mi>T</mi></math></span> has either bounded or unbounded twin-width, and that the growth of <span><math><mi>T</mi></math></span> is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: <span><math><mi>T</mi></math></span> has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs.</div><div>The key for these results is a polynomial time algorithm that takes as input a tournament <span><math><mi>T</mi></math></span> and computes a linear order <span><math><mo><</mo></math></span> on <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> such that the twin-width of the birelation <span><math><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mo><</mo><mo>)</mo></mrow></math></span> is at most some function of the twin-width of <span><math><mi>T</mi></math></span>. Since approximating twin-width can be done in polynomial time for an ordered structure <span><math><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mo><</mo><mo>)</mo></mrow></math></span>, this provides a polynomial time approximation of twin-width for tournaments.</div><div>Our results extend to oriented graphs with stable sets of bounded size, which may also be augmented by arbitrary binary relations.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104247"},"PeriodicalIF":0.9,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145247904","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}

引用次数: 0

Flips in two-dimensional hypertriangulations 二维超三角剖分中的翻转

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-10 DOI: 10.1016/j.ejc.2025.104248

Herbert Edelsbrunner , Alexey Garber , Mohadese Ghafari , Teresa Heiss , Morteza Saghafian

We study flips in hypertriangulations of planar points sets. Here a level-

k

hypertriangulation of

n

points in the plane is a subdivision induced by the projection of a

k

-hypersimplex, which is the convex hull of the barycenters of the

(k - 1)

-dimensional faces of the standard

(n - 1)

-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.

研究平面点集的超三角剖分中的翻转。在这里，平面上n个点的k级超三角剖分是由k-超单纯形的投影引起的细分，k-超单纯形是标准(n - 1)-单纯形的（k−1）维面重心的凸包。特别地，我们引入了四种类型的翻转，并证明了二级超三角剖分是由这些翻转连接起来的。

引用次数: 0

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