填补图兰定理和Pósa猜想之间的空白

On a conjecture of Nikiforov involving a spectral radius condition for a graph to contain all trees

We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Turán type result which might turn out to be of independent interest.

Mubayi的多项式-理想猜想与覆盖理想Turán方法

On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem

本文通过引入覆盖理想方法，反驳了Mubayi关于超图Turán定理的多项式理想猜想，并给出了新的代数证明。

Ramsey 多重性与 Turán 染色

Ramsey multiplicity and the Turán coloring

本文证明 Turán 染色对无限族图是唯一的极值染色，首次确定了 Burr-Rosta 猜想失效图的 Ramsey 多重性常数，并给出条件性的三色结果。

关于Turán定理的两个猜想性强化版本

Two conjectured strengthenings of Turán's theorem

该研究提出两个强化Turán定理的两个谱图论猜想，在多类图上证明猜想成立并给出实验证据，相关猜想被列为谱图论重要未解决问题。

关于禁止构形的Turan（3，4）-问题

On Turan's (3,4)-problem with forbidden configurations

我们确定三个3-图的五个顶点，每个失踪在所有已知的极值配置图兰（3，4）-问题，并证明图兰猜想3-图，另外已知不包含任何诱导副本，这些3-图。我们的论点是基于一个（显然）新的技术“间接解释”，使我们能够检索额外的结构从假设的反例图兰猜想，但在相当松散和有限的意义。我们还包括两个杂项计算的标志代数，证明了类似的结果，其他一些额外的禁止子图。

关于二部图的Turán指数

On Turán exponents of bipartite graphs

本文证明了Erdős-Simonovits猜想对形如$2-2/(2k+1)$的有理数成立，并给出$θ$-图Turán数的上界，从而得到新的有理指数$r=7/5$，推进了相关猜想的研究。

Turan numbers of bipartite subdivisions

Given a graph $H$, the Turán number $ex(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee on the Turán numbers of bipartite graphs, which in turn yields further progress on a conjecture of Erdős and Simonovits. Let $s,t,k\geq 2$ be integers. Let $K_{s,t}^k$ denote the graph obtained from the complete bipartite graph $K_{s,t}$ by replacing each edge $uv$ in it with a path of length $k$ between $u$ and $v$ such that the $st$ replacing paths are internally disjoint. It follows from a general theorem of Buk…

On the rational Turán exponents conjecture

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) = Θ(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of…

离散等距填充循环的上界

An Upper Bound for Discrete Isometric Filling of Cycles

本文研究了离散图度量下循环图C_n的等距填充问题，通过构造同心环形填充，改进了半球上界，证明D*≤1/6。

谱图理论中边缘-谱-图论类型问题的综述：结果、猜想与开放问题

A survey of edge-spectral-Turán type problems in spectral graph theory: Results, conjectures and open problems

本文综述了谱图理论中边缘-谱-图论类型问题的研究成果，提出了相关猜想与开放问题。

广义可加基、Konig引理和Erdos-Turan猜想

Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture

设A是一组非负整数。对于每个非负整数n和正整数h，设r_{A}（n，h）表示n的表示形式n = a_1 + a_2 +. + a_h，其中a_1，a_2，.，a_h是A的元素，a_1 \leq a_2 \leq. \leq a_h.无限集A称为h阶基，如果对每个非负整数n，有r_{A}（n，h）\geq 1. Erdos和Turan证明了对每个2阶基A，limsup_{n\to\infty} r_A（n，2）= \infty.本文引入了一类新的可加基和一般的可加问题，其特例是Erdos-Turan猜想。利用Konig关于某些图中存在无限路的引理，证明了这个一般问题等价于一个有关的非负整数的有限集问题.