  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 $\binom{X}{3} \cup \binom{Y}{3}$ 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.
