The f-Factor Problem for Graphs and the Hereditary Property

$\text{VC}_{\ell}$-dimension and the jump to the fastest speed of a hereditary $\mathcal{L}$-property

In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\text{VC}$-dimension called $\text{VC}_{\ell}$-dimension. Let $\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\mathcal{L}$-property is a class of finite $\mathcal{L}$-structures closed under isomorphism and substructures. The \emph{speed} of a hereditary $\mathcal{L}$-property $\mathcal{H}$ is the function which sends $n$ to $|\mathcal{H}_n|$, where $\mathcal{H}_n$ is the set of elements of $\mathcal{H}$ with universe $\{1,\ldot…

遗传图类的联合嵌入与联合同态的不可判定性

The undecidability of joint embedding and joint homomorphism for hereditary graph classes

本文通过平铺问题归约证明了遗传图类的联合嵌入性质及联合同态性质均为不可判定，为图类理论提供了新的不可判定性结果。

Structure and enumeration theorems for hereditary properties in finite relational languages

Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary $\calL$-properties, where $\calL$ is any finite relational language with maximum arity at le…

图极限（Graphons）与结构极限（Theons）中的弱随机性

Weak randomness in graphons and theons

本文刻画了强持久遗传图族为替换封闭图族，并完全分类了满足弱随机 Erdős--Hajnal 性质的替换封闭图族，即具有少量素图的图族，且将结果推广到有限关系结构。

3-一致超图中的不规则三元组

Irregular triads in 3-uniform hypergraphs

本文研究3-一致超图的驯顺正则分解，刻画线性误差正则分解、齐次性及Fox型正则划分与函数序性质、VC2维数和逐切片VC维数的关系。

Ordered graphs of bounded twin-width

We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs which have unbounded twin-width. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -- under common complexity-theoretic assumptions -- only if the class has bounded twin-width. For hereditary classes of ordered graphs, we show that bounded twin-width is equival…