Abstract
The Turan density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π ∞ (k) consist of all possible Turan densities and let Π fin (k) ⊆ Π ∞ (k) be the set of Turan densities of finite k-graph families. Here we prove that Π fin (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 Π fin (k) contains an irrational number for each k ≥ 3. Also, we show that Π ∞ (k) has cardinality of the continuum. In particular, Π ∞ (k) ≠ Π fin (k) .
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