OPEN
Let $n_1<n_2<\cdots $ be an infinite sequence of integers with associated $a_i\pmod{n_i}$, such that for some $\epsilon>0$ we have $n_k>(1+\epsilon)k\log k$ for all $k$. Then
\[\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).\]
Erdős and Graham
[ErGr80] suggest this 'may have to await improvements in current sieve methods' (of which there have been several since 1980).
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.
