  We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about
the connection of the largest cardinalities (or weight) of structures
satisfying some hereditary property and substructures satisfying the same
hereditary property. We use it to show how results concerning forbidden
subposet problems in the Boolean poset imply analogous results in the poset of
subspaces of a finite vector space. We also study generalized forbidden
subposet problems in the poset of subspaces.
