# Erdős Covering Systems

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set Z. The study of these objects was initiated in 1950 by Erdős [4], and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called “minimum modulus problem” was resolved in 2015 by Hough [6], who proved that in every covering system with distinct moduli, the minimum modulus is at most 1016.
In this mini-course we will give a gentle introduction to the theory of covering systems, and describe some recent progress. In particular, we will give a short proof of Hough’s theorem, and discuss the number of minimal covering systems, the density of the uncovered set. We will also describe a general method for attacking problems about the existence (or otherwise) of covering systems with certain properties.

References
[1]P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, On the Erdős covering problem: the density of the uncovered set, arXiv:1811.03547
[2]P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The Erdős–Selfridge problem with square-free moduli, arXiv:1901.11465
[3]P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806
[4]P. Erdős, On integers of the form 2 k + p and some related problems, Summa Brasil. Math., 2 (1950), 113–123.
[5]M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu, Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc., 20 (2007), 495–517.
[6]R. Hough, Solution of the minimum modulus problem for covering systems, Ann. Math., 181 (2015), 361–382.