Description
Abstract: 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
$f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound.
We show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log
n}}$ 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 $2^{\Omega(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.
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.
https://zoom.us/j/97314411772?pwd=b0kwMVRrQjY3azQ5aUk5MlF2TnRjQT09
Meeting ID: 973 1441 1772
Passcode: 473689