Triangulating real projective n-space

How many vertices do you need to triangulate the real projective n-space? From this blog post of Gil Kalai I learned about a recent preprint (arXiv:2009.02703) by Adiprasito-Avvakumov-Karasev where they construct triangulations with \[\exp\big((1/2 + \mathcal{o}(1))\sqrt{n}\log{n}\big)\text{-many}\] vertices, which is the first construction needing subexponentially-many vertices. More information, also about the history of this problem, may be found in Gil’s blog post.