Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}<p\leq n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p\pmod{p}$.
Estimate $\epsilon_n$ - in particular is it true that $\epsilon_n=o(1)$?
