讲座内容：

For integers i and k, an (i mod k)-cycle is one of length i modulo k. Bollobás proved that for every k and i such that kZ+i contains some even numbers (i.e., k is odd or both k, i are even), there is a smallest number ci,k such that every graph of average degree at least ci,k contains an (i mod k)-cycle as a subgraph. Erdős then asked for the exact value of ci,k. In this talk, we present some results for the extremal graphs without (i mod k)-cycle, for k≤4. Joint work with Yandong Bai, Jun Gao, Ervin Győri, Jie Ma, Yufeng Pan, Nika Salia, Casey Tompkins, Kitti Varga, Tianying Xie, Manran Zhu.