  We study the logical content of several maximality principles related to the
finite intersection principle ($F\IP$) in set theory. Classically, these are
all equivalent to the axiom of choice, but in the context of reverse
mathematics their strengths vary: some are equivalent to $\ACA$ over $\RCA$,
while others are strictly weaker, and incomparable with $\WKL$. We show that
there is a computable instance of $F\IP$ all of whose solutions have
hyperimmune degree, and that every computable instance has a solution in every
nonzero c.e.\ degree. In terms of other weak principles previously studied in
the literature, the former result translates to $F\IP$ implying the omitting
partial types principle ($\mathsf{OPT}$). We also show that, modulo
$\Sigma^0_2$ induction, $F\IP$ lies strictly below the atomic model theorem
($\mathsf{AMT}$).
