Turán numbers of r-graphs on r + 1 vertices

Let $H_k^r$ denote an r-uniform hypergraph with k edges and $r+1$ vertices, where $k\le r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are $\pi \left(H_k^r\right)\le \frac{k-2}{r}$ for all $k\ge 3$, and $\pi \left(H_3^r\right)\ge 2^{1-r}$ for $k=3$. We prove that $\pi \left(H_k^r\right)\ge (C_k-o(1\left)\right)r^{-(1+\frac{1}{k-2})}$ as $r\to \infty$. In the case $k=3$, we prove $\pi \left(H_3^r\right)\ge (1.7215-o(1\left)\right)r^{-2}$ as $r\to \infty$, and $\pi \left(H_3^r\right)\ge r^{-2}$ for all r.