On Possible Turan Densities

2012/04/19 by Pikhurko, Oleg · 1 citation
#05C65 #05D05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1204.4423

Abstract

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≥ 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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