On local Tur\'an problems

Since its formulation, Tur'an’s hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|$\mathcal{F}$| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\left(\genfrac{}{}{0px}{}{X}{3}\right)\cup \left(\genfrac{}{}{0px}{}{Y}{3}\right)$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1,p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.