Exact Turán densities in triple systems

In this paper, we prove several new Turán density results for $3$-graphs. We show: $\pi (C_4^3,\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}{F}_5)=2\sqrt{3}-3$, $\pi (F_{3,2},C_5^{3-})=\frac{2}{9}$, and $\pi (F_{3,2},\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}{F}_{3,2})=\frac{3}{8}$. The first result confirms the conjecture of Shi~[On Turán denisties of small triple graphs, European J. Combin. 52 (2016) 95-102]. The other results give several special non-principal family posed by Mubayi and Rödl~[On the Turán number of triple systems, J. Combin. Theory A. 100 (2002) 135-152].