hypergraph $H$ is a pair $(V,E)$; $V$ is the vertex set, and $E\subseteq 2^V$ is the edge set. If all edges have the same cardinality $k$, then $H$ is a $k$-uniform hypergraph. Turán problems on $k$-uniform hypergraphs have been actively studied for many decades. However, Turán problems on non-uniform hypergraphs are rarely considered (see [@MZ; @Lemons] for two different treatments). Very recently, several groups of people have started actively studying extremal families of sets avoiding given sub-posets. Several new problems have been established. One of them is the diamond problem:

[**The diamond conjecture: [@GriLu]**]{} Any family ${\mathcal{F}}$ of subsets of $[n]:=\{1,2,\ldots,n\}$ with no four sets $A,B,C,D$ satisfying $A\subseteq B\cap C$, $B\cup C \subseteq D$ can have at most $(2+o(1)){\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ subsets.

This conjecture, along with many other problems, motivates us to study Turán-type problems on non-uniform hypergraphs. The details of this connection will be given in the last section.

We briefly review the history of Turán Problems on uniform hypergraphs. Given a positive integer $n$ and a $k$-uniform hypergraph $H$ on $n$ vertices (or $k$-graph, for short), the Turán number ${{\rm ex}}(n,H)$ is the maximum number of edges in a $k$-graph on $n$ vertices that does not contain $H$ as 