Exact codegree Turán density of the complete 3-graph on four vertices

Determine the exact codegree Turán density of the complete 3-uniform hypergraph K_4^{(3)}.

Background

The paper surveys known exact codegree Turán densities for several 3-uniform hypergraphs. It notes that Czygrinow and Nagle conjectured that the codegree Turán density of K_4{(3)} equals 1/2, whereas the analogous conjecture for K_4{(3)-} was subsequently proved using flag algebras. The exact value for K_4{(3)} remains unresolved and is therefore an explicit open problem identified in the paper.

References

Falgas-Ravry, Pikhurko, Vaughan and Volec proved the first conjecture using flag algebras, while the exact value of $\pi_{\mathrm{co}(K_4{(3)})$ remains unknown.

— Any $k$-graph with zero $\ell$-degree Turán density is layered  (2608.18542 - Yang et al., 19 Aug 2026) in Section 1, Introduction