  We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective
modules over a ring $R$, then
$\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair.
Moreover, $\mathfrak{GP}$ is complete when all projective modules are
$\lambda$-pure-injective for some infinite regular cardinal $\lambda$ (in
particular, if $R$ is right $\Sigma$-pure-injective); the latter condition is
shown to be consistent with the axioms of ZFC modulo the existence of strongly
compact cardinals.
  We also thoroughly study $\lambda$-pure-injective modules for an arbitrary
infinite regular cardinal $\lambda$, proving along the way that: any cosyzygy
module in an injective coresolution of a $\lambda$-pure-injective module is
$\lambda$-pure-injective; the cotorsion pair cogenerated by a class of
$\lambda$-pure-injective modules is cogenerated by a set and, under an
additional technical assumption, generated by a set.
  Finally, assuming the set-theoretic hypothesis that $0^\sharp$ does not
exist, we prove that the category of right $R$-modules has enough
$\lambda$-pure-injective objects if and only if the ring $R$ is right
pure-semisimple. This, in turn, follows from a rather surprising result that
$\lambda$-pure-injectivity amounts to pure-injectivity in the absence of
$0^\sharp$.
