Integral points on varieties with infinite étale fundamental group

We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X\mathrm{/}F$ a smooth projective variety, we prove that for any geometrically Galois cover $\phi \text{\hspace{0.17em}⁣}:Y\to X$ of degree at least $2\dim (X{)}^2$, there exists an ample line bundle $\mathcal{L}$ on $Y$ such that for a general member $D$ of the complete linear system $\mathrm{\mid }\mathcal{L}\mathrm{\mid }$, $D$ is geometrically irreducible and any set of $\phi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.