Enumerating Minimal Balanced Collections

Abstract

In this note, 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 at a point other than the origin, and it is called \emph{minimal} if it contains no proper balanced subcollections. We determine the asymptotic number of minimal balanced collections. Specifically, if $B_n$ denotes their total number, then \[ B_n=\frac{2^{n^2-n+1}}{n!}\bigl(1+o(1)\bigr) \qquad\text{as }n\to\infty. \]