Popielarz, Sahasrabuddhe and Snyder in 2018 proved that maximal \(K_{r+1}\)-free graphs with \((1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})\) edges contain a complete r-partite subgraph on \(n-o(n)\) vertices. This was very recently extended to odd cycles in place of \(K_3\) by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.

中文翻译：

---

关于最大无 F 图的稳定性的说明

Popielarz、Sahasrabuddhe 和 Snyder 在 2018 年证明了具有\((1-\frac{1}{r})\frac{n^2}{2}-o 的最大\(K_{r+1}\) -free 图(n^{\frac{r+1}{r}})\)边在\(no(n)\)顶点上包含一个完整的r部分子图。最近，Wang、Wang、Yang 和 Yuan将其扩展到奇数周期代替\(K_3\)。我们进一步将其扩展到其他一些三色图，并在此过程中获得了其他一些稳定性结果。