# The structure and number of Erd\H{o}s covering systems

## TL;DR

This paper determines the asymptotic number of minimal covering systems with a fixed number of elements, providing a structural description of near-optimal systems and resolving a question posed by Erd ext{"o}s in 1952.

## Contribution

It establishes the asymptotic count of minimal covering systems with n elements and offers a structural characterization of systems close to optimal.

## Key findings

- Number of minimal covering systems grows as exp((4√τ/3 + o(1)) n^{3/2}/(log n)^{1/2})
- Provides a structural description of near-optimal covering systems
- Resolves Erd ext{"o}s's 1952 question on the count of minimal covering systems

## Abstract

Introduced by Erd\H{o}s in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erd\H{o}s 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 paper we answer another question of Erd\H{o}s, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly $n$ elements is \[ \exp\left( \left(\frac{4\sqrt{\tau}}{3} + o(1)\right) \frac{n^{3/2}}{(\log n)^{1/2}} \right) \] as $n \to \infty$, where \[ \tau = \sum_{t = 1}^\infty \left( \log \frac{t+1}{t} \right)^2. \] En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense.