Combinatorics of Minimal Balanced Collections

TL;DR

This paper investigates the combinatorial structure of minimal balanced collections, establishing bounds on their quantity and providing insights into their properties within the context of convex geometry and combinatorics.

Contribution

It introduces the concept of minimal balanced collections, defines their properties, and derives upper and lower bounds for their number, advancing understanding in combinatorial convex geometry.

Findings

Established bounds for the number of minimal balanced collections.

Defined the concept of balanced and minimal balanced collections.

Provided asymptotic estimates for their enumeration.

Abstract

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 $\frac{0.288}{n!} \, 2^{(n-1)^2} < B_n < \frac{120}{n!} \, 2^{n^2 - n}$.