Turán's tetrahedron problem asks for the largest number of hyperedges
in a 3-uniform hypergraph on vertices containing no four
vertices with all four possible triples. The conjectured
asymptotic value is
A construction attaining this density partitions the vertices into three nearly equal parts . Its hyperedges are
the triples meeting all three parts and the triples with two vertices
in
and one in ,
with subscripts taken modulo 3 (Keevash 2011).
The ordinary density conjecture remains open. It is distinct from the problem of determining uniform Turán density
(Jeong et al. 2026).
Bloom, T. F. "Erdős Problem 500." Accessed Oct. 26, 2026. https://www.erdosproblems.com/500.Jeong,
G.; Park, S.; Im, S.; Lee, J.; and Yang, H. "A New Upper Bound for the Turán
Density of the Tetrahedron." 23 Sep 2026. https://arxiv.org/abs/2609.27495v2.Keevash,
P. "Hypergraph Turán Problems." In Surveys in Combinatorics 2011
(Ed. R. Chapman). Cambridge, England: Cambridge University Press, pp. 83-140,
2011. https://doi.org/10.1017/CBO9781139004114.004.
of hyperedges
in a 3-uniform hypergraph on 
vertices containing no four
vertices with all four possible triples. The conjectured
asymptotic value is
. Its hyperedges are
the triples meeting all three parts and the triples with two vertices
in 
and one in 
,
with subscripts taken modulo 3 (Keevash 2011).
