Erdos covering systems

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. The study of these objects was initiated by Erdos in 1950, 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, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$.
In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.
Joint work with Paul Balister, Bela Bollobas, Julian Sahasrabudhe and Marius Tiba.