Exchange properties of finite set-systems

TL;DR

This paper investigates the minimal size of specific set-systems with exchange properties, providing bounds that show the existing constructions are nearly optimal and exploring variants with strengthened conditions.

Contribution

The authors establish tight bounds on the size of set-systems satisfying exchange properties, improving understanding of their structure and optimality.

Findings

Lower bound: $f(n) ext{ is at least } 2^{(1.42+o(1))\sqrt{n}}$

Upper bound: $f(n) ext{ is at most } 2^{(1+o(1))\sqrt{2n\log n}}$

For the strengthened variant, bounds are between $2^{\Omega(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$

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 small downward closed set-system $\cal F$ covering an $n$-element ground set which satisfies the following condition: For any two disjoint members $A, B\in\cal F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$ and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and $B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound. It turns out that the construction of Adiprasito et al. is not far from optimal; we show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$. We also study a variant of the above problem, where the condition is strengthened by also requiring that for any two disjoint members $A, B\in\cal F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In this case, we prove that the size of the smallest $\cal F$ satisfying this stronger condition lies between $2^{\Omega(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.