In particular, Baber [@baber] recently found the Turán density of many $3$-uniform hypergraphs using flag algebra methods. For a more complete survey of methods and results on uniform hypergraphs see Peter Keevash’s survey paper [@KeevashSurvey].

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$. Here the edges of $E$ could have different cardinalities. The set of all the cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$. $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. The Lubell function has been a very useful tool in extremal poset theory, in particular it has been used in the study of the diamond conjecture.

In section 2, we show that our notion of $\pi(H)$ is well-defined and is consistent with the usual definition for uniform hypergraphs. We also give examples of Turán densities for several non-uniform hypergraphs. In section 3, we generalize the supersaturation Lemma to non-uniform hypergraphs. Then 