On asymptotic local Turán problems

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and let $t_r(q,p)={\lim }_{n\to \mathrm{\infty }}{t}_r(n,q,p)$. A disjoint union of $k$ complete hypergraphs has $(q,\lceil q\mathrm{/}k\rceil )$-property, which gives $t_r((q,\lceil q\mathrm{/}k\rceil ))\le 1\mathrm{/}{k}^{r-1}$. The first author, Huang and Rödl showed that these constructions are the best asymptotically, that is, ${\lim }_{q\to \mathrm{\infty }}{t}_r((q,\lceil q\mathrm{/}k\rceil ))=1\mathrm{/}{k}^{r-1}$. They asked whether it is true for all real number $\gamma \ge 1$ that ${\lim }_{q\to \mathrm{\infty }}{t}_r((q,\lceil q\mathrm{/}\gamma \rceil ))=1\mathrm{/}\lfloor \gamma {\rfloor }^{r-1}$. In this paper, we give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges.