The purity phenomenon for symmetric separated set-systems

TL;DR

This paper proves that all maximal symmetric separated collections of subsets of [n] have the same size across three separation types, extending known purity results to symmetric set-systems.

Contribution

It introduces the concept of symmetric separated collections and demonstrates a uniform size property (purity) for their maximal instances across three separation types.

Findings

Maximal symmetric separated collections have equal cardinality.

The results extend known purity theorems to symmetric set-systems.

Provides a unified view of purity phenomena in symmetric contexts.

Abstract

Let $n$ be a positive integer. A collection $\cal S$ of subsets of $[n]=\{1,\ldots,n\}$ is called {\it symmetric} if $X\in {\cal S}$ implies $X^\ast\in {\cal S}$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$. We show that in each of the three types of separation relations: {\it strong}, {\it weak} and {\it chord} ones, the following "purity phenomenon" takes place: all inclusion-wise maximal symmetric separated collections in $2^{[n]}$ have the same cardinality. These give "symmetric versions" of well-known results on the purity of usual strongly, weakly and chord separated collections of subsets of $[n]$, and in the case of weak separation, this extends a recent result due to Karpman on the purity of symmetric weakly separated collections in $\binom{[n]}{n/2}$ for $n$ even.