we introduce the area of extremal set theory via three classical results: the erdos-ko rado theorem, frankl's $r$-wise intersection theorem, and the kruskal-katona shadow theorem. we then consider vector space analogs of these problems. we prove a vector space analog of a version of the kruskal-katona theorem due to lov\'{a}sz. we apply this result to extend frank's theorem on $r$-wise intersecting families to vector spaces. in particular, we obtain a short new proof of the erdos-ko-rado theorem for vector spaces.