Zixiang Xu (徐子翔), On the degenerate Turán problems

For a graph $F$, the Turán number is the maximum number of edges in an $n$-vertex simple graph not containing $F$. The celebrated Erdős-Stone-Simonovits Theorem gives that $$\text{ex}(n,F)=(1-\frac{1}{\chi (F)-1}+o(1))(\genfrac{}{}{0ex}{}{n}{2}),$$ where $\chi (F)$ is the chromatic number of $H$. This theorem asymptotically solves the problem when $\chi (F)⩾3$. In case of bipartite graphs $F$, not even the order of magnitude is known in general. In this talk, I will introduce some recent progress on Turán numbers of bipartite graphs and related generalizations and discuss several methods developed in recent years. Finally, I will introduce some interesting open problems on this topic.