Math Conjectures

Turán's Tetrahedron Conjecture

OPENMajorConjectureProposed 1941 · Standard version

Canonical statement

Let K4(3)K_4^{(3)} be the 33-uniform hypergraph consisting of all four triples on a four-element vertex set, and let ex⁡(n,K4(3))\operatorname{ex}(n,K_4^{(3)}) be the largest number of edges in an nn-vertex K4(3)K_4^{(3)}-free 33-uniform hypergraph. Then
lim⁡n→∞ex⁡(n,K4(3))(n3)=59. \lim_{n\to\infty} \frac{\operatorname{ex}(n,K_4^{(3)})}{\binom n3} =\frac59.
View source LaTeX
Let \(K_4^{(3)}\) be the \(3\)-uniform hypergraph consisting
of all four triples on a four-element vertex set, and let
\(\operatorname{ex}(n,K_4^{(3)})\) be the largest number of edges in an
\(n\)-vertex \(K_4^{(3)}\)-free \(3\)-uniform hypergraph. Then
\[
  \lim_{n\to\infty}
  \frac{\operatorname{ex}(n,K_4^{(3)})}{\binom n3}
  =\frac59.
\]
COMB-008:[Turan1941][Keevash2011HypergraphTuran][Razborov2010K43]
Combinatorics

Notes

Write K4(3)K_4^{(3)} for the complete 33-uniform hypergraph on four vertices, the tetrahedron. Turán's conjecture of 1941 asserts that the largest K4(3)K_4^{(3)}-free 33-uniform hypergraph on nn vertices has edge density tending to 5/95/9. It is the oldest and most prominent instance of the hypergraph Turán problem, posed in the same paper in which Turán settled the corresponding question for graphs completely [Turan1941].

The lower bound comes from Turán's constructions achieving density 5/95/9, and a notorious feature of the problem is that many essentially different constructions attain this same density, which obstructs the usual stability approaches; see Keevash's survey of the area [Keevash2011HypergraphTuran]. In the other direction, Razborov's flag-algebra method gives the best known upper bounds, only slightly above the target, at about 0.5616660.561666 [Razborov2010K43].

The gap between 5/9≈0.55565/9\approx0.5556 and roughly 0.56170.5617 has resisted all subsequent refinements, and proving that the limiting density equals 5/95/9 exactly remains open.

References (3)

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