On Possible Turan Densities

The Turán density π(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 Turán densities and let F^k be the set of Turán 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.

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