On the algebraic and topological structure of the set of Turán densities

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• 作者：Codruţ Grosu¹ ; ^{grosu.codrut@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Tur&aacute ; n densities ; Uniform hypergraphs ; Jumps
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：118
• 期：Complete
• 页码：137-185
• 全文大小：789 K

文摘

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$r\ge 3$. 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.