An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the upper bound on the smallest modulus to $616000$.
The best known lower bound is a covering system whose minimum modulus is $42$, due to Owens [Ow14].
Hough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].
Selfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).
The answer is no, as proved by Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], who (among other results) prove that if \[1< C \leq N^{\frac{\log\log\log N}{4\log\log N}}\] then, for any $N\leq n_1<\cdots< n_k\leq CN$, the density of integers not covered for any fixed choice of residue classes is at least \[\prod_{i}(1-1/n_i)\] (and this density is achieved for some choice of residue classes as above).
These bounds were improved by Croot [Cr03b] who proved \[\frac{N}{L(N)^{\sqrt{2}+o(1)}}< f(N)<\frac{N}{L(N)^{1/6-o(1)}},\] where $f(N)=\exp(\sqrt{\log N\log\log N})$. These bounds were further improved by Chen [Ch05] and then by de la Bretéche, Ford, and Vandehey [BFV13] to \[\frac{N}{L(N)^{1+o(1)}}<f(N) < \frac{N}{L(N)^{\sqrt{3}/2+o(1)}}.\] The latter authors conjecture that the lower bound here is the truth.
That is, for all $d\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is congruent to some $a_d\pmod{d}$, and if there is some integer $x$ with \[x\equiv a_d\pmod{d}\textrm{ and }x\equiv a_{d'}\pmod{d'}\] then $(d,d')=1$.
Adenwalla [Ad25] has proved there are no such $n$.
In general, for any $n$ one can try to choose such $a_d$ to maximise the density of integers so covered, and ask what this density is. This was also investigated by Adenwalla [Ad25].
Is the minimum density achieved when all the $a_i$ are equal?
For $k=1$ or $k=2$ any set $A$ such that $\sum_{n\in A}\frac{1}{n}=\infty$ has this property.
Note that since the $k$th prime is $\sim k\log k$ the lower bound $n_k>(1+\epsilon)k\log k$ is best possible here.
Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?