  An old question of Erdos asks if there exists, for each number N, a finite
set S of integers greater than N and residue classes r(n) mod n for n in S
whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is
bounded for such a covering of the integers, then the least member of S is also
bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a
conjecture of Erdos and Graham, that, for each fixed number K>1, the complement
in the integers of any union of residue classes r(n) mod n, for distinct n in
(N,KN], has density at least d_K for N sufficiently large. Here d_K is a
positive number depending only on K. Either of these new results implies
another conjecture of Erdos and Graham, that if S is a finite set of moduli
greater than N, with a choice for residue classes r(n) mod n for n in S which
covers the integers, then the largest member of S cannot be O(N). We further
obtain stronger forms of these results and establish other information,
including an improvement of a related theorem of Haight.
