Maximal independent sets and separating covers

TL;DR

This paper explores the maximum size of separating covers and their connection to maximal independent sets in graphs, revealing a surprising relationship with the factorization of integers and providing combinatorial insights.

Contribution

It establishes a novel combinatorial explanation linking separating covers, maximal independent sets, and integer factorization, extending Moon and Moser's results.

Findings

Maximum number of subsets in a separating cover equals the largest product of positive integers summing to n

Provides a combinatorial explanation connecting separating covers and graph theory

Shows how these results relate to the complexity of integers

Abstract

In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns out to be the inverse to the answer to a much simpler question: what is the largest integer which is the product of positive integers with sum n? We give a combinatorial explanation for this relationship, via Moon and Moser's answer to a question of Erdos: how many maximal independent sets can a graph on n vertices have? We conclude by showing how Moon and Moser's solution also sheds light on a problem of Mahler and Popken's about the complexity of integers.