# The Turán density of tight cycles in three-uniform hypergraphs

# The Turán density of tight cycles in three-uniform hypergraphs

**In-Person Talk **

Turán-type problems for hypergraphs have been an intriguing area of research. Despite significant efforts, the Turán density of F is known for only a few three-uniform hypergraphs F. This talk concerns Turán-type problems for 3-uniform tight cycles C_k, where the number of vertices k is not divisible by 3. The Turán density of a hypergraph F is the maximum density of an n-vertex hypergraph that does not contain any member of F. Mubayi and Rödl gave an ``iterated blow-up'' construction showing that the Turán density of C_5 is at least 2sqrt{3}-3, and this bound is conjectured to be tight. Interestingly, their construction also excludes C_k for larger k not divisible by 3, indicating that it might be the extremal construction for these hypergraphs as well. Indeed, we have recently shown that the Turán density of C_k is 2sqrt{3}-3 for sufficiently large k, in a joint result with Shoham Letzter and Alexey Pokrovskiy.