  We offer the following explanation of the statement of the Kuratowski graph
planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas
intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to
every cell there corresponds a unique cell with the complementary set of
vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere;
there exist precisely two 3-dimensional dichotomial cell complexes, and their
1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their
1-skeleta all but one graphs of the Petersen family.
  In higher dimensions n>2, we observe that in order to characterize those
compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited
minors", it suffices to establish finiteness of the list of all (n-1)-connected
n-dimensional finite cell complexes that do not embed in S^{2n} yet all their
proper subcomplexes and proper cell-like combinatorial quotients embed there.
Our main result is that this list contains the n-skeleta of (2n+1)-dimensional
dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell
complexes include (apart from the three joins of the i-skeleta of
(2i+2)-simplices) at least ten non-simplicial complexes.
