ON possible Turán densities

By Oleg Pikhurko. 2012

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The Turán 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 Turán densities and let Πfin(k) ⊆ Π∞(k) be the set of Turán densities of finite k-graph families. Here we prove that Πfin(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 Πfin(k) contains an irrational number for each k ≥ 3. Also, we show that Π∞(k) has cardinality of the continuum. In particular, Π∞(k) ≠ Πfin(k).

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2541c2f0ab6103ff10da0ad5aa239913721d1ef3