On a system of equations with primes

Given an integer $n\ge 3$, let $u_1,\dots ,u_n$ be pairwise coprime integers $\ge 2$, $\mathcal{D}$ a family of nonempty proper subsets of ${1, \ldots, n}$with\mathrm{"}enough\mathrm{"}elements,and$\varepsilon$afunction$ \mathcal D \to {\pm 1}$\mathrm{.}Doesthereexistatleastoneprime$q$suchthat$q$divides$\prod_{i \in I} u_i - \varepsilon(I)$forsome$I \in \mathcal D$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $\mathcal{D}$ are subjected to certain restrictions. We use the result to prove that, if $\varepsilon_0 \in {\pm 1}$and$A$ is a set of three or more primes that contains all prime divisors of any number of the form ${\prod }_{p\in B}p-{\epsilon }_0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.