I was wondering if there are results for Turan densities of 4- or 5-graphs (hypergraphs). I am aware of several surveys of hypergraph Turan densities (due to Pikhurko, Keevash), but these mainly focus on 3-graphs.
I am only aware of several results for 4- or 5- graphs (or higher-order hypergraphs in general); apologies in advance if I misattribute any results:
Markstrom has shown the Turan density in graphs forbidding $K_{5}^{4}$ is upper bounded by ~0.736. Sidorenko has shown a lower bound of $\frac{11}{16}$, which is conjectured to be exact.
de Caen and Sidorenko have shown that the Turan $(K_{r+1}^{r})^{-}$-density in graphs forbidding $K_{r+1}^{r}$ is upper bounded by $1 - \frac{1}{r}$ (here I use $(K_{r+1}^{r})^{-}$ to denote $K_{r+1}^{r}$ minus an edge). (I think de Caen also has a general upper and lower bound for $K_{s}^{r}$.) Lu and Zhao tighten this density in the case where $r$ is even.
But what I am looking for is Turan densities for the "stranger cases". I know for 3-graphs there are lower bounds for many weird Turan densities, like $\pi(K_{4}^{-}, C_{5})$, or $\pi(K_{4}^{-}, C_{5}, F_{3,2})$ (and here I am mostly referring to the results in this paper). Is there anything similar for 4- or 5-graphs?