Erdős–Sós question on the uniform Turán density of K4^(3)

Determine whether the uniform Turán density of the tetrahedron satisfies $\pi_u\!\left(K_4^{(3)}\right)=\frac{1}{2}$ . Here $K_4^{(3)}$ is the complete $3$ -uniform hypergraph on $4$ vertices, and $\pi_u(H)$ is the supremum of the real numbers $d$ such that, for every $\varepsilon>0$ and all sufficiently large $n$ , there exists an $H$ -free $3$ -uniform hypergraph $G$ on $n$ vertices for which every subset $U\subseteq V(G)$ with $|U|\geq\varepsilon n$ satisfies $e(G[U])\geq(d-\varepsilon)\binom{|U|}{3}$ .

References

Primary source

arXiv

Additional references

The uniform Turán density of the tetrahedron — arXiv — Matija Bucić

extremal-combinatorics hypergraph-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

An unrefereed September 2026 preprint claims the exact answer is one half, changing the problem from open to apparently solved but not yet independently confirmed.

The Erdős–Sós question asks whether the uniform Turán density of $K_4^{(3)}$ equals $1/2$ . Until this announcement, the question was explicitly described as open, with $1/2$ believed optimal.

Known results

Rödl’s construction gives the lower bound $1/2$ for $K_4^{(3)}$ .
A 2024 palette theorem reduces uniform Turán densities to palette constructions, but does not establish the $K_4^{(3)}$ case.

September 2026 claimed solution

Matija Bucić’s arXiv preprint The uniform Turán density of the tetrahedron claims that the exact density is $1/2$ , resolving the Erdős–Sós question. The result is reported as unrefereed and remains unverified.

Current status (as of September 2026): The value $1/2$ is claimed by a new unrefereed preprint, but independent verification is not recorded; the earlier lower bound and conjectural status are established.

Sources

No solutions have been posted yet.