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 \[ B_n=\frac{2^{n^2-n+1}}{n!}\bigl(1+o(1)\bigr) \qquad\text{as }n\to\infty. \]
From: Mikhail Bludov V [view email]
[v1]
Mon, 24 Nov 2025 17:17:19 UTC (28 KB)
[v2]
Thu, 23 Jul 2026 11:03:12 UTC (47 KB)
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