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

A system of congruences

$$a_i \pmod{n_i}, \qquad n_1 < n_2 < \cdots < n_k$$

is called a covering system if every integer satisfies at least one of the congruences in (3). The simplest covering system is $0 \pmod 2$, $0 \pmod 3$, $1 \pmod 4$, $5 \pmod 6$, $7 \pmod{12}$. The main problem is: Is it true that for every $c$ one can find a covering system all whose moduli are larger than $c$? I offer 1000 dollars for a proof or disproof.