Uniform Turán density beyond 3-graphs

In the 1980s, Erdős and Sós first introduced an extremal problem on hypergraphs with density constraints. Given an r-uniform hypergraph F (or r-graph for short), its uniform Turán density πᵤ(F) is the smallest value of d in which every hypergraph H in which every linear-sized subhypergraph of H has edge density at least d contains F as a subgraph. The first non-zero value of πᵤ(F) was not found until 30 years later. Progress in studying the set of values of the uniform Turán density of r-graphs has been uneven in terms of r: to this day there are infinitely many non-zero values known for $r=3$, a single non-zero value known for $r=4$ and none for r≥ 5. In this paper we obtain the first explicit values of πᵤ for all uniformities, by proving that for every r≥ 3 there exist r-graphs F with πᵤ(F)=1/4 and with πᵤ(F)=r2^-r2.