one additional vertex, $\ast$, added to every edge of $H$. In a hypergraph Turán problem workshop hosted by the AIM Research Conference Center in 2011, the following conjecture was posed: $\displaystyle \lim_{t\to\infty}\pi(S^t(K^{r}_n))=0$. We conjecture $\displaystyle\lim_{t\to\infty} \pi(S^t(H))=|R(H)|-1$ holds for any hypergraph $H$. Some partial results are proved. Finally in the last section, we will point out the relation between the Turán problems on hypergraphs and extremal poset problems.

Non-uniform hypergraphs
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Notation
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Recall that a hypergraph $H$ is a pair $(V,E)$ with the vertex set $V$ and edge set $E\subseteq 2^{V}$. Here we place no restriction on the cardinalities of edges. The set $R(H):=\{|F|\colon F\in E\}$ is called the set of its [*edge types*]{}. A hypergraph $H$ is $k$-uniform if $R(H)=\{k\}$. It is non-uniform if it has at least two edge types. For any $k\in R(H)$, the [*level hypergraph*]{} $H^k$ is the hypergraph consisting of all $k$-edges of $H$. A uniform hypergraph $H$ has only one (non-empty) level graph, i.e., $H$ itself. In general, a non-uniform hypergraph $H$ has $|R(H)|$ (non-empty) level hypergraphs. Throughout the paper, for any finite set $R$ of non-negative integers, we say, $G$ is an $R$-graph if $R(G)\subseteq R$. We write $G^R_n$ for a hypergraph on 