Erdős Problem #2 — Covering systems with arbitrarily large minimum modulus

Let $a_i \pmod{n_i}$, with $n_1<n_2<\cdots<n_k$, be a finite system of congruences. Call it a covering system if every integer satisfies at least one of the congruences. Is it true that, for every $c$, there exists a covering system whose moduli all exceed $c$, equivalently, with $n_1>c$?