4 problems

Let $r\geq 2$ be an integer, and let $\Pi_{fin}^{r}$ denote the set of Turán densities of finite $r$-uniform hypergraphs. Chung–Graham's rationality conjecture. Every element of…

Small-length Turán density conjecture. Both equalities should already hold with $\ell=5$; equivalently, the asserted formulas are conjectured for every $\ell\geq 5$ not divisible b…

Extremal construction conjecture. The iterated blow-up of a hyperedge is the extremal construction for $\mathcal{C}^-_5$ and attains the conjectured Turán density…

For integers $k\geq 2$, let $K_{k+1}^k$ be the complete $k$-graph on $k+1$ vertices. For a $k$-graph $H$, let $\delta_{k-1}(H)$ be its minimum $(k-1)$-degree, and let…