On Possible Turan Densities
  The Tur\'an density \pi(H) of a family H of k-graphs is the limit as n tends to infinity of the maximum edge density of an H-free k-graph on n vertices. Let I^k consist of all possible Tur\'an densities and let F^k be the set of Tur\'an densities of finite k-graph families.   Here we prove that F^k 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 F^k contains an irrational number for each k\ge 3.   Also, we show that I^k has cardinality of the continuum. In particular, I^k is not equal to F^k. 
Oleg Pikhurko
math.CO