As usual, for an $r$-uniform hypergraph $G$, denote by $ex\_r(n,G)$ the maximum number of edges an $r$-uniform, $G$-free hypergraph on $n$ vertices can have, and let $\lim \frac{ex\_r(n,G)}{\binom nr}\stackrel{n\to\infty}\longrightarrow\pi\_r(G)$.  

Define the maximum of this quantity over graphs with $m$ edges as $\pi\_r(m)=\max \{\pi\_r(G) : e(G)=m\}$.



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> Is it true that $\lim \pi\_r(m)\stackrel{r\to\infty}\longrightarrow 0$ for every $m$?
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For $m=1,2$ this follows from well-known statements for set families, but I couldn't find anything for higher values.  

On one hand, this feels like it should be simple, on the other, it seems to have strong consequences.



Turán density of hypergraphs with very few edges
Seems, the answer is negative for $m=3$: consider $H = \{AB, BC, AC\}$ for disjoint $k$-element sets $A,B,C$. Then for $r=2k$ the is an $r$-graph $F$ which density is close to 0.5: consider only $r$-tuples that have an odd intersection with the first half-part $V\_1$ of vertex set. Suppose there is a copy of $H$. Then at least two of sets $A,B,C$ have intersection with $V\_1$ of the same parity. Union of two such sets do not belong to $F$; a contradiction.


