# The weak separation in higher dimensions

## TL;DR

This paper introduces and explores the properties of weakly r-separated collections of subsets in higher dimensions, extending prior concepts and providing a geometric framework for their combinatorial analysis.

## Contribution

It generalizes weak separation to higher dimensions and develops a geometric approach to analyze maximal collections, extending previous work by Leclerc and Zelevinsky.

## Key findings

- Established combinatorial properties of maximal weakly r-separated collections
- Extended weak separation concepts to higher dimensions and even r cases
- Developed a geometric framework for analyzing these collections

## Abstract

For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and Zelevinsky. In this paper, extending results due to Leclerc-Zelevinsky, we develop a geometric approach to establish a number of nice combinatorial properties of maximal weakly r-separated collections. As a supplement, we also discuss an analogous concept when $r$ is even.