  Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We
prove that, if $\mathcal{S}$ has the $(\omega, 2)$-property (for any infinitely
many sets in $\mathcal{S}$, at least $2$ among them intersect), then
$\mathcal{S}$ can be partitioned into finitely many subfamilies, each with the
finite intersection property. If $\mathcal{S}$ is definable in some first-order
structure, then these subfamilies can be chosen definable too. This is a
strengthening of the case $q=2$ of the definable $(p,q)$- conjecture in model
theory and of the Alon-Kleitman-Matou\v{s}ek $(p,q)$-theorem in combinatorics.
