What is the minimum number of n-simplexes needed to triangulate the n-cube

Definitions and mathematical statement

Let $Q_n=[0,1]^n$ and let its vertex set be $\{0,1\}^n$. A vertex triangulation of $Q_n$ is a finite collection of nondegenerate $n$-simplices whose vertices belong to $\{0,1\}^n$, whose union is $Q_n$, and such that the intersection of any two simplices is either empty or a common face of both. In particular, their interiors are disjoint. Lower-dimensional faces are included implicitly in the resulting simplicial complex.

Define the simplexity in this convention by \[ t(n)=\min\{\text{number of full-dimensional simplices in such a triangulation}\}. \] Determine $t(n)$, or obtain asymptotically sharp bounds as $n\to\infty$. The usual triangulation obtained by ordering the coordinates shows $t(n)\le n!$, but it need not be optimal.

The corner restriction excludes added Steiner vertices, and the common-face condition excludes arbitrary simplicial dissections with incompatible subdivisions across interfaces. These are different minimization problems, even when useful bounds compare them. Counting refers only to $n$-dimensional simplices, not to all faces in the complex. An upper bound requires an actual compatible triangulation covering the cube, while a lower bound must apply to every such triangulation, irrespective of symmetry or regularity.

Short English statement

Determine the minimum number of n-simplices in a triangulation of the n-cube using only cube vertices.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100009 (AMR-030-0009), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Geometry, source-order bullet 9. Original source indexed by AMR-030-0009. https://people.math.sc.edu/cooper/combprob.html.
• S3 Hughes, Simplexity of the cube, definitions of triangulation and simplexity. https://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/Hughes/Hughes6.pdf.