Erdős Problem #27 — Density covered by congruences with moduli between $t$ and $ct$

Let us now restrict ourselves to a special case. The $n_i$ are the integers between $t$ and $ct$. First of all denote by $\alpha_1(c,t)$ the smallest possible value of the density of the integers satisfying none of the congruences $a_i \pmod{m}$ ($t \leq m \leq ct$), and let $\alpha_2(c,t)$ be the largest possible value of the density of the integers satisfying none of these congruences. Now as we already remarked $\alpha_1(c,t)$ can be clearly made at least as large as $1 - \prod_{t \leq u \leq ct}(1 - 1/u)$. Can it in fact be made much larger? It is easy to see that it can be 1 only if there is a covering congruence the smallest modulus of which is $\geq t$ and the largest modulus of which is $ct$. On the other hand perhaps there is a $c$ so that for every $\varepsilon > 0$ there is a $t$ for which $\alpha_1(c,t) > 1 - \varepsilon$. I am not at all sure if this is possible and I give 100 dollars for an answer.