Local $t$-dimension

On scattered posets with finite dimension

We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.

Boxicity and Poset Dimension

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ is the ground set and $P$ is a reflexive, anti-symmetric and…

单环偏序集的维数

Dimension of unicycle posets

受随机偏序集维数研究的启发，Bollobás和Brightwell于1997年猜想：若有限偏序集的覆盖图至多含一个环，则其序维数至多为3。本文通过构造性证明给出显式的线性扩展三元组，证实该猜想。

Subdividing the cd-index of Eulerian Posets

This thesis aims to give the reader an introduction and overview of the cd-index of a poset, as well as establish some new results. We give a combinatorial proof of Ehrenborg and Karu's cd-index subdivision decomposition for Gorenstein* complexes and extend it to a wider class of subdivisions. In doing so, we define a local cd-index that behaves analogously to the well studied local h-vector. We examine known cd-index and h-vector bounds, and then use the local cd-index to bound a particular class of polytopes with the cd-index of a stacked polytope. We conclude by investigating the h-vector a…

Adjacency posets of outerplanar graphs

Felsner, Li and Trotter showed that the dimension of the adjacency poset of an outerplanar graph is at most 5, and gave an example of an outerplanar graph whose adjacency poset has dimension 4. We improve their upper bound to 4, which is then best possible.

The complexity of counting poset and permutation patterns

We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for uncons…

关于偏序集乘积维度的一些令人沮丧的问题

Some frustrating questions on dimensions of products of posets

研究有限维偏序集乘积的维度函数行为，探讨其与单独偏序集维度之和的关系，并提出关于维度问题的新旧问题及相关概念。

On the dimension of posets with cover graphs of treewidth $2$

In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Biró, Keller, and Young). This can be interpreted as evidence that the dimension sho…

Partial-conjugates and Dimensionality of Posets

The Pareto dominance relation of a preference profile is (the asymmetric part of) a partial order. For any integer n, the problem of the existence of an n-agent preference profile that generates the given Pareto dominance relation is to investigate the dimension of the partial order. We provide a characterization of a partial order having dimension n in general.

Posets with cover graph of pathwidth two have bounded dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant $d$ such that if $P$ is a poset with cover graph of $P$ of pathwidth at most $2$, then $\dim(P)\leq d$. We answer this question in the affirmative by showing that $d=17$ is sufficient. We also show that if $P$ is a poset containing the standard example $S_5$ as a subposet, then the cover graph of $P$ has treewidth at least $3$.

平面性与维度I

Planarity and dimension I

本文证明了具有平面覆盖图的偏序集的维度为$\mathcal{O}(s^8)$，其中$s$是该偏序集中最大标准例子的阶数。

Improved lower bounds on the on-line chain partitioning of posets of bounded dimension

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounde…

Boxicity, poset dimension, and excluded minors

In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.

Incidence Posets and Cover Graphs

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of Kříž and Nešetřil, we show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two.

数值半群、多面体与偏序集 III：极小表示与面维度

Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension

作为Kunz多面体$P_m$组合学系列研究的第三篇，本文刻画数值半群极小表示的Kunz偏序集可恢复性，证明同面内部半群的极小表示基数一致，并给出由Kunz偏序集求面维度的组合方法。

On Subdivision Posets of Cyclic Polytopes

There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$.

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this inequality is tight when $P$ contains the standard example $S_n$. However, there are posets with large dimension that do not contain the standard example $S_2$. Moreover, for each fixed $d\ge2$, if $P$ is a poset with $|P|\le 2n+1$ and $P$ does not contain the standard example $S_d$, then $\dim(P)=o(n)$. Also, for large $n$, there is a poset $P$ with $|P|=2…

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset $P$, the Boolean dimension of $P$ and the local dimension of $P$ are both bounded from above by the dimension of $P$ and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height…

Ext and local cohomology modules of face rings of simplicial posets

There are a large number of theorems detailing the homological properties of the Stanley--Reisner ring of a simplicial complex. Here we attempt to generalize some of these results to the case of a simplicial poset. By investigating the combinatorics of certain modules associated with the face ring of a simplicial poset from a topological viewpoint, we extend some results of Miyazaki and Gräbe to a wider setting.

Local Dimension is Unbounded for Planar Posets

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension …

三维偏序集的Kelly–Trotter乘积猜想

The Kelly--Trotter product conjecture for posets of dimension three

该研究证明了三维偏序集下的Kelly–Trotter乘积猜想，否定了Trotter的相关猜想，还得出特定偏序集乘积维数的结论，所用方法涉及偏序集分类与图着色搜索。

可除性序与多重集偏序的维数

The Dimension of Divisibility Orders and Multiset Posets

本文研究偏序集的Dushnik-Miller维数，改进了整数区间上可除性序的维数上界至O((log κ)^3/(log log κ)^2)，并推广到多重集偏序及多项式可除性序。

递归定义偏序集的序与链多面体的单纯形不等式

Simplex inequalities of order and chain polytopes of recursively defined posets

本文研究了有限偏序集的序多面体和链多面体的单纯形面，证明由$\mathbf{X}$-free偏序集递归构造的偏序集，其链多面体在任意维度的单纯形面数量不少于序多面体，推广了Mori的先前结果。

关于二维 Cohen-Macaulay 偏序集与置换图

On Cohen-Macaulay posets of dimension two and permutation graphs

本文刻画了二维 Cohen-Macaulay 偏序集为可壳化且强连通的偏序集，并利用余可比图与置换图的关系，给出了 Cohen-Macaulay 置换图的特征描述。

A local method for posets

We propose some conditions on a poset that produce a small chain complex for its homology. This allows to compare simplicial complexes and Quillen's complexes under the same prism. It turns out they differ in the existence or not of free faces in an acyclic complex.

可切换性空间的维数

Dimensions of toggleability spaces

本文证明了 Defant 等人关于若干偏序集可切换性空间维数的猜想，并将其推广到受限图定义的一类更广偏序集，证明维数等于偏序集的秩加一。

Dimension is polynomial in height for posets with planar cover graphs

We show that height $h$ posets that have planar cover graphs have dimension $\mathcal{O}(h^6)$. Previously, the best upper bound was $2^{\mathcal{O}(h^3)}$. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes $K_5$ as a minor.

Planar posets have dimension at most linear in their height

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets of height $h$ and dimension at least $(4/3)h-2$.

关于偏序集维度的进一步思考

Further thoughts on dimensions of posets

本文围绕有限偏序集维度展开，回顾相关概念与猜想，指出两个可推出该猜想的不等式，还定义了\(P\preccurlyeq P'\)和\(P\approx P'\)关系，给出关于这些关系的结果及问题。

Kelly--Trotter猜想与偏序集乘积的维数

The Kelly--Trotter conjecture and dimension of poset products

本文研究有限偏序集乘积的序维数，否证了Kelly--Trotter猜想，构造了具有$(3,d)$-覆盖性质的偏序集$Q_d$，并给出$(r,s)$-覆盖的等价刻画。

超平面排列中区域的偏序的序维数

The Order Dimension of the Poset of Regions in a Hyperplane Arrangement

研究了Coxeter群弱序的序维数等于其秩，并为其他有限类型给出了序维数的界限，通过统一方法简化了A和B类型已知结果。

偏序集维数与最大可比度

Poset dimension and maximum comparability degree

本文证明了有限偏序集的维数上界可改进为$O(d \log d)$，其中$d$为可比图最大度，从而确认了Erdős等人下界的紧性。

Garland's Technique for Posets and High Dimensional Grassmannian Expanders

Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannia…

平面性与维度II

Planarity and dimension II

研究具有平面（哈斯）图的偏序集维度计算问题，给出常数因子多项式时间近似算法，通过对标准示例的结构理解得出算法结果，证明了\(\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672\)这一更强界。

On difference graphs and the local dimension of posets

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is $n$ while its local dimension is only $3$. Hiraguchi (1955) proved that the maximum dimension of a poset of order $n$ is $n/2$. However, we find a very different result for local dimension, proving a bound of $Θ(n/…

Random bipartite posets and extremal problems

Previously, Erdős, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0<p\le 1/2$. While bounds were given for the range $1/2\le p <1$, the relative accuracy of the results in the original paper deteriorated as $p$ approa…

On an extremal problem for poset dimension

Let $f(n)$ be the largest integer such that every poset on $n$ elements has a $2$-dimensional subposet on $f(n)$ elements. What is the asymptotics of $f(n)$? It is easy to see that $f(n)\geqslant n^{1/2}$. We improve the best known upper bound and show $f(n)=\mathcal{O}(n^{2/3})$. For higher dimensions, we show $f_d(n)=\mathcal{O}\left(n^\frac{d}{d+1}\right)$, where $f_d(n)$ is the largest integer such that every poset on $n$ elements has a $d$-dimensional subposet on $f_d(n)$ elements.

通过少量映射导航偏序集

Navigating Posets with Few Maps

本文研究了偏序集的两个新参数，旨在高效确定给定元素的后继集。通过定义平面映射和紧元素的概念，将mapability和atlas thickness与经典维度和宽度联系起来，并展示了计算这两个参数的复杂性。

Dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\…

团宽与维数

Cliquewidth and dimension

本文证明有界团宽且维数足够大的偏序集必含标准例子或Kelly例子，并据此完整刻画了覆盖图在子图封闭图类中时偏序集维数有界的条件，同时证明有界团宽蕴含布尔维数有界。

Boolean dimension and local dimension

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x<y$?". This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a v…

The proof of the removable pair conjecture for fractional dimension

In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. This paper settles this latter conjecture.

A Note on Covering Young Diagrams with Applications to Local Dimension of Posets

We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning any Young diagram with $\binom{2k}{k}-1$ steps into actual rectangles, each row and each column used by at most $k$ rectangles. This answers two questions by Kim et al. (2018). Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of p…

2-Dimension from the topological viewpoint

In this paper we study the 2-dimension of a finite poset from the topological point of view. We use homotopy theory of finite topological spaces and the concept of a beat point to improve the classical results on 2-dimension, giving a more complete answer to the problem of all possible 2-dimensions of an n-point poset.

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'ı}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant l…

Fractional Local Dimension

The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has been studied extensively in the literature. In 1992, Brightwell and Scheinerman developed the notion of fractional dimension as the natural linear programming relaxation of the Dushnik-Miller concept. In 2016, Ueckerdt introduced the concept of local dimension, and in just three years, several research papers studying this new parameter have been published. In this paper, we introduce and study fractional local dimension. As suggested by the terminology, our parameter is a common generalization of …

Additive posets, CW-complexes, and graphs

We introduce and study additive posets. We show that the top homology group (with coefficients in Z/2Z) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.

Topological minors of cover graphs and dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result …

Algebraic properties of classes of path ideals

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen-Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise from the Luce-decomposable model in algebraic statistics, can be viewed as path ideals of certain posets. We study invariants of these so-called \emph{Luce-decomposable} monomial ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their regularity and t…

Boolean Dimension, Components and Blocks

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if $\dim(C)\le d$ for every component $C$ of a poset $P$, then $\dim(P)\le \max\{2,d\}$; also if $\dim(B)\le d$ for every block $B$ of a poset $P$, then $\dim(P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if $\text{ldim}(C)\le d$ for every component $C$ of a poset $P$, then $\text{ldim}(P)\le d+2$; however, for every $d\ge 4$, there exists a poset $P$ with…

偏序集中的局部对称性

Local symmetries in partially ordered sets

本文通过偏序集自同构引入局部对称性，构造更易枚举的对称类，并用于区分离散时空模型中的因果集。

有限偏序集到诺特UFD谱的保维饱和嵌入

Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs

本文刻画了有限偏序集可保维饱和嵌入局部诺特UFD谱的充分必要条件，并进一步构造了任意有限偏序集到半局部拟优环谱的饱和嵌入，使其极小元对应极小素理想且余高与最长链长度一致。

Minors and dimension

It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et …

偏序集上层的共收缩扩张及其在优质2查询局部可测码与提升码中的应用

Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes

本研究提出偏序集上层的共收缩扩张局部判据，将其应用于局部可测码领域，证明优质2查询LTCs的存在性并给出提升码局部可测性的纯局部判据，改进了此前需混合局部全局条件的结果。