Exact Tur\'{a}n densities in triple systems

In this paper, we prove several new Tur\'{a}n density results for $3$-graphs. We show: $\pi(C_4^3, \mathrm{complement\ of\ } F_5) = 2\sqrt{3} - 3$, $\pi(F_{3,2}, C_5^{3-}) = \frac{2}{9}$, and $\pi(F_{3,2}, \mathrm{induced\ complement\ of\ } F_{3,2}) = \frac{3}{8}$. The first result confirms the conjecture of Shi~[On Tur\'an 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 R\"odl~[On the Tur\'an number of triple systems, J. Combin. Theory A. 100 (2002) 135-152].