A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a $2$-colouring of the edges of a large complete graph is asymptotically minimized by a random colouring. It is well known that the disjoint union of two common graphs may be uncommon; e.g., $K_2$ and $K_3$ are common, but their disjoint union is not. We investigate the commonality of disjoint unions of multiple copies of $K_3$ and $K_2$. As a consequence of our results, we obtain an example of a pair of uncommon graphs whose disjoint union is common. Our approach is to reduce the problem of showing that certain disconnected graphs are common to a constrained optimization problem in which the constraints are derived from supersaturation bounds related to Razborov's Triangle Density Theorem. We also improve bounds on the Ramsey multiplicity constant of a triangle with a pendant edge and the disjoint union of $K_3$ and $K_2$.
Unified approach to the generalized Turán problem and supersaturation
本文介绍了图论中广义Turán问题和过饱和结果的一种统一方法。过饱和极值函数$satex(n, F : m, G)$是一个$n$ -顶点图所能拥有的子图$G$的最少拷贝数，它至少包含$F$的$m$拷贝作为子图。我们提出了一个调查，讨论以前已知的结果，并获得几个新的主要集中在证明方法，极值结构和相变现象。最后指出了与超图极值问题的关系，特别是Berge-type结果。
Spectral supersaturation for color-critical graphs
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Supersaturation of induced even cycles in locally sparse graphs
Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）；EPFL（洛桑联邦理工学院）；University of Oxford（牛津大学）；École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；School of Mathematics and Statistics, Henan Normal University（河南师范大学数学与统计学院）
Supersaturation in Nosal graphs: Triangles and books
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
Supersaturation thresholds for expanded color-critical graphs
Fuzhou University（福州大学）；Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；Key Laboratory for Operations Research and Cybernetics of Fujian Universities（福建省高校运筹与控制重点实验室）
An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学中心）；Yau Mathematical Sciences Center (YMSC), Tsinghua University（清华大学丘成桐数学科学中心）
Asymptotic Density of Graphs Excluding Disconnected Minors
对于图$H$，设$$c_{\infty}(H)= \lim_{n \to \infty}\max\frac{|E(G)|}{n},$$，其中最大值取在不包含$H$作为子图的$n$顶点上的所有图$G$上。因此，$c_{\infty}(H)$是不包含$H$作为子图的图的渐近最大密度。利用Eppstein的一个结构引理，证明了不连通图$H$的$c_{\infty}(H)$的新上界.特别地，只要$H$是圈的并集，我们就确定$c_{\infty}(H)$。最后，我们研究了固定$r$的$c_\infty(sK_r)$的行为，其中$sK_r$表示完全图的$s$不相交副本在$r$顶点上的并集。本文改进了Pakason的结果，证明了$$c_\infty(sK_r)=s(r-1)-1 \mathrm{\; for \;} s =Ω\left(\frac{\log{r}}{\log\log{r}}\right),$$和$$c_\infty(sK_r)>s(r-1)-1 \mathrm{\; for \;} s ={o}\left(\frac{\log{r}}{\log\log{r}}\right).$$
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；School of Mathematics, Hunan University（湖南大学数学学院）
Disconnected graphs and extremal bounds for realizable distance orders
Instituto de Matemáticas, UNAM（墨西哥国立自治大学数学研究所）；Facultad de Ciencias, UNAM（墨西哥国立自治大学理学院）；Escuela Nacional de Estudios Superiores, UNAM（墨西哥国立自治大学高等研究国立学院）；Centro de Ciencias Matemáticas, UNAM（墨西哥国立自治大学数学科学中心）
