# Rob Morris "Erdős covering systems"

**Rob Morris ** from IMPA will
give the talk "Erdős covering systems" on the labs' Big Seminar.

Password: first 6 decimal places of $\pi$ after the decimal point

You can also write to Alexander Polyanskii (alexander.polyanskii@yandex.ru) or to Maksim Zhukovskii (zhukmax@gmail.com) if you want to be added to mailing list.

**Abstract:**

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 Erdős 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, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

## Watch the video:

Everyone is invited to attend. The language of the lecture is English. The event is aimed at master and graduate students, as well as researchers in the field of combinatorics.