Hypergraph Turan Problem

The Turan function ex(n,F) of a k-graph F is the maximum
number of edges in an F-free k-graph on n vertices. This problem goes
back to the fundamental paper of Turan from 1941 that solved it for
complete graphs (k=2). Unfortunately, very few non-trivial instances of
the problem have been solved when we consider hypergraphs (k>2).

We survey some recent results and methods on the hypergraph Turan
function. In particular, we discuss the so-called stability approach
that greatly helps in obtaining exact results from asymptotic
computations (for example, those that use flag algebras or graph
limits).