  Reiher, R\"odl and Schacht [J. London Math. Soc. 97 (2018), 77--97] showed
that the uniform Tur\'an density of every $3$-uniform hypergraph is either $0$
or at least $1/27$, and asked whether there exist $3$-uniform hypergraphs with
uniform Tur\'an density equal or arbitrarily close to $1/27$. We construct
$3$-uniform hypergraphs with uniform Tur\'an density equal to $1/27$.

Combinatorics