Exchange properties of finite set-systems

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 f(n) the
smallest cardinality of such a family, they proved that
f(n)<2O(n√logn), and they asked for a nontrivial lower bound.
We show that 2(1.42+o(1))n√≤f(n)≤2(1+o(1))2nlogn√ 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 Ω(n√logn) and O(nloglogn/logn).
It remains an interesting open problem to reduce this gap.

To warm-up, you can solve this problem:
https://www.komal.hu/feladat?a=feladat&f=A797

Joint work with Peter Frankl and Janos Pach.