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Erdős Covering Systems

Published online by Cambridge University Press:  23 May 2024

By
Paul Balister
Edited by
Felix Fischer  and
Robert Johnson
Felix Fischer
Affiliation:
Queen Mary University of London
Robert Johnson
Affiliation:
Queen Mary University of London
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Summary

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of infinite arithmetic progressions whose union is the set of all integers. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until the last decade little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most a fixed constant. The ideas of Hough were simplified and extended in 2018 by Balister, Bollobas, Morris, Sahasrabudhe and Tiba, to give solutions (or progress towards solutions) to a number of related questions. We give a summary of this and other progress that has been made since.

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Chapter
Information
Surveys in Combinatorics 2024 , pp. 31 - 54
Publisher: Cambridge University Press
Print publication year: 2024

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