  In this expository paper we present short simple proofs of
Conway-Gordon-Sachs' theorem on intrinsic linking in three-dimensional space,
as well as van Kampen-Flores' and Ummel's theorems on intrinsic intersections.
The latter are related to nonrealizability of certain hypergraphs in
four-dimensional space. The proofs use a reduction to lower dimensions which
allows to exhibit relation between these results. We use elementary language
which allows to present the main ideas without technicalities. Thus our
exposition is accessible to non-specialists in the area, including students who
know basic three-dimensional geometry, and who are ready to learn
straightforward four-dimensional generalizations.
