Thm. Given a monomial ideal, the vanishing set is a union of coordinate subspaces.
Proof: take one of the monomial generators, and look at the variables appearing in it. One of them must vanish. So V(I) is a union over those generators x_i, of the vanishing set of a monomial ideal on the coordinate hyperplane {x_i=0}. By induction, inside each one of those the vanishing set is a union of coordinate subspaces.
We defined simplicial complexes. This page about them is more about topology, but might be easier to understand. They correspond 1:1 to squarefree monomial ideals, as explained here (though that has a lot of extra stuff we haven't talked about).
We defined prime ideals, and showed that any intersection of prime ideals (such as just one) is radical. We looked at the prime ideals in Z, C[x], and R[x].
