A "Challenging Question" of Björner from 1976: Every Infinite Geometric Lattice of Finite Rank Has a Matching

A Note on Slice Rank and Matchings in Groups

A multiplicative 3-matching in a group $G$ is a triple of sets $\{a_i\}, \{b_i\}, \{c_i\} \subset G$ such that $a_ib_jc_k = 1$ if and only if $i=j=k$. Here we record the fact that $\text{PSL}(2,p)$ has no multiplicative 3-matching of size greater than $O(p^{8/3})$, yet the slice rank of its group algebra's multiplication tensor is at least $Ω(p^3)$ over any field. This gives a negative answer to a conjecture of Petrov.

相邻纸牌匹配问题

Matching adjacent cards

该研究针对标准52张牌组及k花色n点数的广义牌组，解答了同点数相邻牌的概率、平均匹配数及匹配数概率分布问题，还确定了k固定n趋于无穷时的极限分布。

完美匹配、Fano平面与$E_8$型正交基

Perfect matchings, Fano planes, and orthogonal bases of type $E_8$

利用完美匹配和标记Fano平面构造并研究$E_8$根系中的2025个正交基，这些基构成一个Bruhat型偏序集，通过广义Rothe图的基数计算秩函数，并给出组合刻画与秩生成函数。

The rank of a complex unit gain graph in terms of the matching number

A complex unit gain graph (or ${\mathbb T}$-gain graph) is a triple $Φ=(G, {\mathbb T}, φ)$ (or $(G, φ)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, φ)$, the set of unit complex numbers $\mathbb{T}= \{ z \in C:|z|=1 \}$ and a gain function $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $φ(e_{i,j})=φ(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \leq r(G, φ) \leq 2m(G)+c(G)$, where $r(G, φ)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, φ)$, the matching number and the cyclomatic number of $G$, respect…

Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms

We develop a theory of graph algebras over general fields. This is modeled after the theory developed by Freedman, Lovász and Schrijver in [22] for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex a…

关于二分图中的$P$-顶点问题

The Full P-vertex Problem and Perfect Matchings for Bipartite Graphs

研究二分图中性质(P)与完美匹配的关系，证明性质(P)等价于完美匹配存在的条件，并应用于多类二分图族。

Domination in intersecting hypergraphs

A matching in a hypergraph $H$ is a set of pairwise disjoint hyperedges. The matching number $α'(H)$ of $H$ is the size of a maximum matching in $H$. A subset $D$ of vertices of $H$ is a dominating set of $H$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $H$. The cardinality of a minimum dominating set of $H$ is called the domination number of $H$, denoted by $γ(H)$. It is known that for a intersecting hypergraph $H$ with rank $r$, $γ(H)\leq r-1$. In this paper we present structural properties on intersecting hypergraphs with rank $r$ satis…

模阿贝尔独立群中的匹配与乘积增长

Matchings and product growth in modular abelian independence groups

该研究统一了群有限子集与域扩张有限维子空间的两种匹配理论，引入独立群概念，推导模阿贝尔独立群的匹配准则，并得到推广Cauchy–Davenport定理的乘积增长界。

随机奖励、产品排名与未知耐心下的在线匹配框架

Online Matching Frameworks under Stochastic Rewards, Product Ranking, and Unknown Patience

研究随机奖励、产品排名与未知耐心下的在线匹配问题，提出基于资源消耗的框架，在多种到达模型下推导新结果，并给出级联点击模型下的最优与近似算法及新模型下的0.027近似算法。