# The Erd\H{o}s-Selfridge problem with square-free moduli

## TL;DR

This paper investigates the Erdős-Selfridge problem on covering systems with distinct moduli, proving that square-free moduli cannot all be odd, thus establishing the necessity of an even modulus under these conditions.

## Contribution

It proves that in covering systems with square-free, distinct moduli, at least one modulus must be even, resolving a special case of a long-standing open problem.

## Key findings

- Square-free, distinct moduli covering systems must include an even modulus.
- Addresses a special case of Erdős's question on odd moduli.
- Provides new constraints on the structure of covering systems.

## Abstract

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erd\H{o}s in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erd\H{o}s, asks whether there exist covering systems whose moduli are distinct and all odd. We show that if in addition one assumes the moduli are square-free, then there must be an even modulus.