Conditions forcing the existence of relative complements in lattices and posets

局部几何偏序集的Orlik–Solomon代数与复阿贝尔排列的上同调

The Orlik--Solomon algebra of a locally geometric poset and cohomology of complex abelian arrangements

该研究构造了局部几何偏序集的Orlik–Solomon代数，利用其与复阿贝尔李群子群排列的关联，结合拓扑信息计算排列补集的有理上同调，尤其给出紧李群情形下的显式微分分次代数。

Arrangements of Spheres and Projective Spaces

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections an…

Zero loci of nullvectors and skew zero forcing in graphs and hypergraphs

There is interesting internal structure in the nullspaces of graph and hypergraph adjacency matrices, especially for trees, bipartite graphs, and related combinatorial classes. The zero loci of nullvectors, i.e., their zero coordinates' indices, encode information about matchings, coverings, and edges' influence on rank. This set system is the lattice of flats of a ``kernel matroid'', a subsystem of which are the ``stalled'' sets closed under skew zero forcing (SZF), a graph percolation/infection model known to have connections with rank and nullity. For a wide variety of graphs, the lattice o…

On Zero Forcing Number of Graphs and Their Complements

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G) \setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank -- Special Graphs Work Group". It's known that $Z(G)\geq δ(G)$, where $δ(G)$ is the minimum degree of $G$. We show that $Z(G)\leq n…

莱默码与避免(32-1)模式的排列到避免(3-21)模式的排列的逆补映射

Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations

该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。

陪集偏序集的壳

Shelling the Coset Poset

证明了有限群的陪集格有可壳序复形当且仅当该群是可补的。此外，陪集格被证明有一个Cohen-Macaulay序复形在完全相同的条件。所使用的群论工具是相对初级的，并且避免了有限单群和最小有限单群的分类。

Twin-Width is Linear in the Poset Width

Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomasse and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation…

Two Examples of Toric Arrangements

We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).

有限偏序集的迭代性质

Iterative properties of birational rowmotion

本文研究了有限偏序集的birational rowmotion映射，证明其在(p,q)-矩形偏序集上有有限阶，并探讨了其在三角形和skeletal类偏序集上的性质，揭示了其与Zamolodchikov周期猜想的类比关系。

在有限维偏序的(不可比)图的极小支配集计数

Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets

本文提出在有限维偏序的不可比图中，通过flashlight搜索方法实现输出多项式时间算法，并改进翻转方法以减少空间需求，从而提升极小支配集的枚举效率。

A Conjecture of Kozlov from the 1998 Proceedings of the American Mathematical Society: Non-Evasive Order Complexes and Generalizations of Non-Complemented Lattices

Let $P$ be a finite poset with an element $s$ such that (1) for all $x\in P$, either $s\vee x$ or $s\wedge x$ exists; and (2) for all $x,y\in P$ such that $x<y$, if $s\wedge x$ does not exist but $s\wedge y$ does exist, then $(s\wedge y)\vee x$ exists. Kozlov, the winner of the 2005 European Prize in Combinatorics ("for deep combinatorial results obtained by algebraic topology and particularly for the solution of a conjecture of Lovász"), conjectured in the 1998 Proceedings of the American Mathematical Society that the order complex of $P$ is non-evasive. We prove this conjecture.

Forcing a sparse minor

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this question has been studied in depth when $H$ is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined $f(H)$ when $H$ has a super-linear number of edges. We focus on the case when $H$ has a linear number of edges. Our main result, which complements the result of Myers and Thomason…