Abstract
The present paper is concerned with the various algebraic structures supported by the set of Turán densities.We prove that the set of Turán densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a consequence we construct explicit irrational densities for any r≥3. The proof relies on a technique recently developed by Pikhurko.We also show that the set of all Turán densities forms a graded ring, and from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs.Finally, we prove that the set of Turán densities of families of r-graphs has positive Lebesgue measure if and only if it contains an open interval. This is a simple consequence of Steinhaus's theorem.
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