Verify the Turán density conjecture for 3-uniform hypergraphs

Prove or disprove the conjecture that the Turán density t(s,3) equals 4/(s−1)^2 for every integer s≥4.

Background

The paper identifies a proposed general formula for the Turán density of complete s-vertex subhypergraphs in 3-uniform hypergraphs. Several constructions are known to attain the value 4/(s−1)2, but the statement is described as a conjecture rather than an established theorem. Resolving it would determine t(s,3) for all s≥4.

References

Tur\n{a}n and other researchers conjectured that $t(s,3) = \frac{4}{(s-1)2}$ for $s \ge 4$.

— A note on the minimum size of Turán systems  (2501.15457 - Liu et al., 26 Jan 2025) in Section 1, Introduction