The Turán density of tight cycles in three-uniform hypergraphs (Q6623562)

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 \(n\rightarrow \infty\). Denote by \(\mathcal{C}_{\ell}\) the 3-uniform tight cycle on \(\ell\) vertices. \textit{D. Mubayi} and \textit{V. Rödl} [J. Comb. Theory, Ser. A 100, No. 1, 136--152 (2002; Zbl 1009.05074)] 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. In this paper, the Turán density of \(\mathcal{C}_{\ell}\) for all large \(\ell\) not divisible by 3 is found, showing that indeed \(\pi(\mathcal{C}_{\ell})=2\sqrt{3}-3\). In the opinion of the authors, 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 their 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''. Some open problems conclude the paper.