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^-r{2}.
Ander Lamaison (Thu,) studied this question.