Turan problems on Non-uniform Hypergraphs

A non-uniform hypergraph H = (V, E) consists of a vertex set V and an edge set E subset of 2(V); the edges in E are not required to all have the same cardinality. The at of all cardinalities of edges in II is denoted by R(II), the set of edge types. For a fixed hypergraph H, the Turan density pi(H) is defined to be lim(n ->infinity) max(Gn) h(n)(G(n)), where the maximum is taken over all H-free hypergraphs G(n) on n vertices satisfying R(G(n)) subset of R(H). and h(n)(G(n)), the so called Lubell function, is the expected number of edges in (In hit by a random full chain. This concept, which generalizes the Turan density of k-uniform hypergraphs, is motivated by recent work on extremal poset problems. The details connecting these two areas will be revealed in the end of this paper. Several properties of Turan density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turan densities of {1, 2}-hypergraphs.