In a recent breakthrough, Adiprasito, Avvakumov, and Karasev
constructed a triangulation of the n-dimensional real projective space
with a subexponential number of vertices. They reduced the problem to
finding a set-system satisfying certain properties. Denoting by the
smallest cardinality of such a family, they proved that
, and they asked for a nontrivial lower bound.
We show that for such families.
We also study a variant of the above problem, where we prove that the
size of the smallest family satisfying a slightly stronger condition
lies between and .
It remains an interesting open problem to reduce this gap.
To warm-up, you can solve this problem:
Joint work with Peter Frankl and Janos Pach.