Combinatorics of Minimal Balanced Collections

TL;DR

The paper investigates the enumeration of minimal balanced collections of subsets of $[n]$, linking these combinatorial objects to full-rank $0$-$1$ matrices and weight vectors via geometric interpretations of convex hulls and the cube. It builds a framework around weight-vector unificators and a $\mathbb{Z}_2$-action to bound the count $B_n$ of minimal balanced collections, proving asymptotically sharp upper and lower bounds of order $2^{n^2}/n!$, with a refined bound $B_{n,n} \ge \frac{\alpha}{n!} 2^{(n-1)^2}$ where $\alpha\approx 0.288$. The main contributions are the precise connection between minimal balanced collections and positive full-rank matrices, the weight-vector counting approach, and the use of symmetry and Sperner-type arguments to control the growth of $B_n$. These results advance understanding of core feasibility in cooperative games and provide a method that could extend to higher-subset sizes and other finite fields.

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

Theorems & Definitions (53)

• Definition 1.1
• Theorem 1
• Definition 2.1
• Definition 2.2
• Proposition 3.1
• proof
• Lemma 3.1
• Definition 4.1
• Lemma 4.1
• proof