Tur7an density of the complete 3-uniform hypergraph on five vertices

Determine the exact Tur7an density \(\pi(K^{(3)}_5)\) of the complete 3-uniform hypergraph on five vertices.

Background

The paper proves that the Tur7an basis density for rank-3 matroids with no U_{3,5}-restriction is $\pi_M(3,U_{3,5})=3/4$. Via the correspondence between uniform matroid minors and daisy subgraphs, this confirms Tur7an's conjectured value in the restricted class of matroid basis hypergraphs.

The unrestricted hypergraph problem is substantially harder: the relevant forbidden configuration is $K^{(3)}_5$, the complete 3-uniform hypergraph on five vertices. Although the paper records a construction giving the lower bound 3/4, it explicitly states that the unrestricted Tur7an density remains unresolved.