Combinatorics of Minimal Balanced Collections

In this article, we explore the combinatorics of balanced collections. A collection of subsets of the set $[n] = \{1, \dots, n\}$ is called \emph{balanced} if the relative interior of the convex hull of the corresponding characteristic vectors intersects the main diagonal of the $n$-dimensional cube, and it is called \emph{minimal} if it contains no proper balanced subcollections. In particular, we establish both upper and lower bounds for the number of minimal balanced collections. Specifically, we prove that if $B_n$ denotes the number of minimal balanced collections, then $\frac{0.288}{n!} \, 2^{(n-1)^2} < B_n < \frac{120}{n!} \, 2^{n^2 - n}$.