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Since their introduction by Erd\H{o}s in 1950, covering systems (that is,finite collections of arithmetic progressions that cover the integers) havebeen extensively studied, and numerous questions and conjectures have beenposed regarding the existence of covering systems with various properties. Inparticular, Erd\H{o}s asked if the moduli can be distinct and all arbitrarilylarge, Erd\H{o}s and Selfridge asked if the moduli can be distinct and all odd,and Schinzel conjectured that in any covering system there exists a pair ofmoduli, one of which divides the other.  Another beautiful conjecture, proposed by Erd\H{o}s and Graham in 1980,states that if the moduli are distinct elements of the interval $[n,Cn]$, and$n$ is sufficiently large, then the density of integers uncovered by the unionis bounded below by a constant (depending only on $C$). This conjecture wasconfirmed (in a strong form) by Filaseta, Ford, Konyagin, Pomerance and Yu in2007, who moreover asked whether the same conclusion holds if the moduli aredistinct and sufficiently large, and $\sum_{i=1}^k \frac{1}{d_i} < C$. Althoughthis condition turns out not to be sufficiently strong to imply the desiredconclusion, as the main result of this paper we will give an essentially bestpossible condition which is sufficient.  Our method has a number of further applications. Most importantly, we provethe conjecture of Schinzel stated above, which was made in 1967. We moreovergive an alternative (somewhat simpler) proof of a breakthrough result of Hough,who resolved Erd\H{o}s' minimum modulus problem, with an improved bound on thesmallest difference. Finally, we make further progress on the problem ofErd\H{o}s and Selfridge.

On the Erd\H{o}s Covering Problem: the density of the uncovered set