Turán's tetrahedron problem

How many triples can be chosen from $n$ points without ever covering all four triples of some 4-point set, a "tetrahedron"? Turán found a construction using a $5/9$ fraction of all triples and conjectured that it is optimal. This is the simplest unsolved case of the hypergraph Turán problem, and Erdős offered $500 for it.

Problem

A 3-graph is a set of triples from an $n$ -element vertex set. Let $K_4^{3}$ denote the complete 3-graph on four vertices (all four triples of a 4-set), and let $\mathrm{ex}_3(n,K_4^3)$ be the maximum number of triples in a 3-graph on $n$ vertices that contains no copy of $K_4^3$ . Determine $\mathrm{ex}_3(n,K_4^3)$ ; in particular, is the Turán density

\pi(K_4^3)=\lim_{n\to\infty}\frac{\mathrm{ex}_3(n,K_4^3)}{\binom n3}

equal to $5/9$ ?

Background

The problem is due to Turán and is the natural extension of his theorem on graphs to 3-graphs. His construction splits the vertices into three equal parts $X_1,X_2,X_3$ and takes every triple with one vertex in each part, together with every triple having two vertices in $X_i$ and one in $X_{i+1}$ (indices modulo 3). It contains no $K_4^3$ and has $(5/9+o(1))\binom n3$ triples. Erdős listed the problem among those he most wanted to see solved and offered $500 for it. No Turán density $\pi(K_r^{s})$ of a complete $s$ -graph with $r>s>2$ has been determined, and the tetrahedron is the first case; Keevash's survey gives the wider context.

What is known

Lower bound. Turán's construction gives $\pi(K_4^3)\ge 5/9\approx0.5556$ .
Upper bounds. The best bounds come from Razborov's flag-algebra method. Razborov (2010) reported that numerical flag-algebra computations suggest $\pi(K_4^3)\le0.561666$ (the source page misprints this as $0.5611666$ ), and Baber (2012) proved $0.5615$ using vertex-coloured flags with regularity constraints. An August 2026 forum post reads off $0.561465$ from Baber's own certificate, checked in exact rational arithmetic but not yet published.
A solved variant. Erdős and Sós asked for the Turán density when the triples must be spread uniformly over the host. For the tetrahedron this uniform Turán density was shown to equal $1/2$ in September 2026, independently by Kielak, Král', Lamaison, Liu, Shu and Wu and by Bucić (preprints). The ordinary density asked for here is unaffected.

What would count as progress

A smaller upper bound on $\pi(K_4^3)$ , delivered as a flag-algebra certificate (positive semidefinite matrices with exact rational entries) that a script can check.
A $K_4^3$ -free 3-graph with more than $(5/9+\varepsilon)\binom n3$ triples for large $n$ , which would disprove the conjecture; such an example is checked by enumerating 4-sets.
A proof that $\pi(K_4^3)=5/9$ , or a stability result describing all near-extremal configurations.
A formal proof against the Lean statement in formal-conjectures.

References

P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42. doi:10.1007/BF02579174
A. A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010), 946–963. doi:10.1137/090747476
R. Baber, Turán densities of hypercubes (2012). arXiv:1201.3587
P. Keevash, Hypergraph Turán problems, in Surveys in Combinatorics 2011, Cambridge University Press (2011), 83–140. doi:10.1017/CBO9781139004114.004
B. Kielak, D. Král', A. Lamaison, H. Liu, X. Shu and Z. Wu, Solution of uniform Turán's tetrahedron problem (2026). arXiv:2609.08336

Source

Imported from Erdős Problems 500, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.