Determine the Turán density for uniformity at least three

Determine the exact value of the Turán density t(s,r) for at least one pair of integers s>r≥3, thereby resolving the general exact-value problem for Turán systems.

Background

For integers n≥s>r, the quantity T(n,s,r) is the minimum number of r-edges in an n-vertex hypergraph such that every s-vertex set contains an edge. Its limiting normalized value is the Turán density t(s,r). The paper notes that the exact value is known for graphs (r=2) through Turán’s theorem, but not for higher uniformities. Determining even one exact value for s>r≥3 is presented as a longstanding problem in extremal combinatorics, with a historical monetary incentive offered by Erdős.