Frick–Hosseini–Vasileuski triangulation lower-bound question

The question concerns lower bounds for the number of vertices in triangulations of real projective space. Earlier work established subexponential upper bounds and asked for a nontrivial lower bound.

Known results

• Adiprasito, Avvakumov, and Karasev (2022) constructed triangulations with a subexponential number of vertices and reduced the lower-bound problem to a downward-closed set-system question.
• For every positive integer $n$, an earlier construction gives at most $e^{(\frac12+o(1)\sqrt n\log n}$ vertices for a triangulation of $\mathbb{R}P^{n-1}$.

September 2026 asymptotic lower bound

On September 9, 2026, Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel–Lindenstrauss–Milman theorem claimed the first superpolynomial lower bound and an asymptotic determination of the minimum up to $\exp(d^{o(1)})$ in the exponent. The preprint is unrefereed.

Current status (as of September 2026): A claimed unrefereed preprint provides the first superpolynomial lower bound and asymptotic resolution, but the result remains unverified.