讲座内容：

Erd\H{o}s and Simonovits (1973) proposed the following problem: For an integer $r\geq 2$ and a family $\mathcal{F}$ of non-bipartite graphs, what is the maximum of minimum degree $\delta(G)$ among all $n$-vertex $\mathcal{F}$-free graphs $G$ with chromatic number at least $r$? The edge and spectral counterpart of this problem is to determine the maximum size and spectral radius $\lambda (G)$ among all $n$-vertex $\mathcal{F}$-free graphs $G$ with chromatic number at least $r$. In this talk, we will first introduce some relevant background and then present some of our extremal results for $C_{2k+1}$-free graphs with high chromatic number.