ON POSSIBLE TURAN DENSITIES

The Turan density pi(T) of a family F of k-graphs is the limit as n -> infinity of the maximum edge density of an F-free k-graph on n vertices. Let Pi((k))(infinity) consist of all possible Turan densities and let Pi((k))(fin) subset of Pi((k))(infinity) be the set of Turan densities of finite k-graph families. Here we prove that Pi((k))(fin) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Pi((k))(fin) contains an irrational number for each k >= 3. Also, we show that Pi((k))(infinity) has cardinality of the continuum. In particular, Pi((k))(infinity) not equal Pi((k))(fin).