The Turán Density of Tight Cycles in Three-Uniform Hypergraphs

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”

The codegree threshold of K4−K_4^-

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$ vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration $G$, there is a quasirandom tournament $T$ on the same vertex set such that $G$ is close in the edit distance to the $3$-graph $C(T)$ whose edges are the cyclically oriented triangles from $T$. For infinitely many values of $n$, we are further able to determine $\mathrm{ex}_2(n, K_4^-)$ exactly and to show that tournament-based constructions $C(T)$ are extremal for those values of $n$.Comment: 31 pages, 7 figures. Ancillary files to the submission contain the information needed to verify the flag algebra computation in Lemma 2.8. Expands on the 2017 conference paper of the same name by the same authors (Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413