The Turán density of tight cycles in three-uniform hypergraphs

Abstract

The Turán density of an $r$-uniform hypergraph $\mathcal{H}$, denoted $\pi (\mathcal{H})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n\to \infty$. Denote by ${\mathcal{C}}_{\ell }$ the $3$-uniform tight cycle on $\ell$ vertices. Mubayi and Rödl gave an ``iterated blow-up'' construction showing that the Turán density of ${\mathcal{C}}_5$ is at least $2\sqrt{3}-3\approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain ${\mathcal{C}}_{\ell }$ for larger $\ell$ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of ${\mathcal{C}}_{\ell }$ for all large $\ell$ not divisible by $3$, showing that indeed $\pi ({\mathcal{C}}_{\ell })=2\sqrt{3}-3$. To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement ``a graph is bipartite if and only if it does not contain an odd cycle''.

34 pages, 4 figures (final version accepted to IMRN plus a few comments in conclusion)

Cited by in corpus (1)

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