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arXiv · 1901.11465

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

Paul Balister·Béla Bollobás·Robert Morris·Julian Sahasrabudhe·Marius Tiba
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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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Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, Marius Tiba. 2019-01-31. The Erd\H{o}s-Selfridge problem with square-free moduli. https://doi.org/10.2140/ant.2021.15.609

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