On September 9, 2026, Florian Frick, Kaave Hosseini, Eric Myzelev, Arya Narnapatti and Aliaksei Vasileuski posted a preprint (Frick et al. 2026) which gives a superpolynomial lower bound for the number of vertices needed to triangulate real projective space. Together with constructions found only a few years ago, their theorem essentially determines the asymptotic answer to a long-standing problem in combinatorial topology.
Recall that the -dimensional real projective space is the set of all lines through the origin in . Equivalently, consider the unit sphere
Every line through the origin meets in two opposite points and , so can be obtained from by identifying every point with its antipodal point .
For example, is homeomorphic to a circle. The space is the usual real projective plane.
A finite simplicial complex consists of a finite set of vertices and a collection of finite subsets of these vertices, called simplices, such that every subset of a simplex is again a simplex. A simplex with two vertices is an edge, one with three vertices is a triangle, one with four vertices is a tetrahedron, and so on. By gluing geometric simplices according to this combinatorial information, one obtains the geometric realization of the complex.
A simplicial triangulation of a topological space is a finite simplicial complex whose geometric realization is homeomorphic to . Thus, informally, the problem is to build by gluing together simplices.
A natural measure of the complexity of such a triangulation is its number of vertices. Let denote the smallest possible number of vertices in a simplicial triangulation of .
How quickly does grow with ?
There is an elementary-looking construction with exponentially many vertices. Start with a suitable antipodally symmetric triangulation of and identify opposite points. In particular, a classical construction gives
up to an inessential shift of the dimension.
For a long time no construction with substantially fewer vertices was known. On the other hand, the best general lower bounds were only polynomial. In particular, Arnoux and Marin proved that, for ,
(Arnoux and Marin 1991).
Thus there was an enormous gap: perhaps the correct answer was polynomial in , perhaps exponential, or perhaps somewhere in between.
A breakthrough came from Karim Adiprasito, Sergey Avvakumov and Roman Karasev. In a paper published in 2022, they constructed triangulations of with
vertices (Adiprasito et al. 2022). This was the first construction to break the exponential barrier.
Their argument was surprisingly short. They first constructed a centrally symmetric polytope whose boundary has an antipodal symmetry with special combinatorial properties, and then took its quotient by this symmetry to obtain a triangulation of projective space.
The combinatorial part of their construction was subsequently improved by Peter Frankl, J’anos Pach and D"om"ot"or P’alv"olgyi (Frankl et al. 2022). Combining the two works gives
This showed that the answer is subexponential. But it left open the opposite question: could there be dramatically smaller triangulations, perhaps even ones with only polynomially many vertices?
The new theorem rules this out.
Here the notation
means that there is an absolute constant such that, for all sufficiently large ,
Together with the known upper bound, we therefore have
Since
this can be summarized by the remarkably precise formula
Thus the new theorem determines the correct power of in the exponent. There remains a factor of order between the best known upper and lower bounds for , but the main asymptotic question is settled: the minimum number of vertices is much larger than every polynomial in , yet much smaller than .