The Erd\H{o}s-Selfridge problem with square-free moduli
Number Theory
2021-05-26 v1 Combinatorics
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.
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Cite
@article{arxiv.1901.11465,
title = {The Erd\H{o}s-Selfridge problem with square-free moduli},
author = {Paul Balister and Béla Bollobás and Robert Morris and Julian Sahasrabudhe and Marius Tiba},
journal= {arXiv preprint arXiv:1901.11465},
year = {2021}
}
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17 pages