Georgia Institute of Technology College of Sciences

Hypergraph Turán Problems became more approachable due to flag algebras. In this talk we will first focus on tight cycles without an edge. A tight $k$-cycle minus an edge $C_k^-$ is the 3-graph on the vertex set $[k]$, where any three consecutive vertices in the string $123...k1$ form an edge. We show that for every $k \geq 5$, k not divisible by $3$, the extremal density is $1/4$. Moreover, we determine the extremal graph up to $O(n)$ edge edits. The proof is based on flag algebra calculations.

Then we describe new developments in solving large semidefinite programs that allows for improving several other bounds on Turán densities.

This talk is based on joint work with Connor Mattes, Florian Pfender and Jan Volec.