Vol. 15, No. 3, 2021

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The Erdős–Selfridge problem with square-free moduli

Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba

Vol. 15 (2021), No. 3, 609–626

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 Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, 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.

covering systems, Erdős–Selfridge problem
Mathematical Subject Classification 2010
Primary: 11B25
Secondary: 11A07, 11N35
Received: 31 January 2019
Revised: 14 August 2020
Accepted: 18 September 2020
Published: 20 May 2021
Paul Balister
University of Oxford
United Kingdom
Béla Bollobás
University of Cambridge
United Kingdom
University of Memphis
Memphis, TN
United States
Robert Morris
Rio de Janeiro
Julian Sahasrabudhe
University of Cambridge
United Kingdom
Marius Tiba
University of Cambridge
United Kingdom