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Enochs Conjecture asserts that each covering class of modules (over any ring)has to be closed under direct limits. Although various special cases of theconjecture have been verified, the conjecture remains open in its fullgenerality. In this paper, we prove the conjecture for the classes$\mathrm{Filt}(\mathcal S)$ where $\mathcal S$ consists of $\aleph_n$-presentedmodules for some fixed $n<\omega$. In particular, this applies to the left-handclass of any cotorsion pair generated by a class of $\aleph_n$-presentedmodules.  Moreover, we also show that it is consistent with ZFC that Enochs Conjectureholds for all classes of the form $\mathrm{Filt}(\mathcal{S})$ where$\mathcal{S}$ is a set of modules. This leaves us with no explicit example of acovering class where we cannot prove that the Enochs Conjecture holds (possiblyunder some additional set-theoretic assumption).

Enochs Conjecture for cotorsion pairs and more