A long standing conjecture of Bagemihl (1956) states that there
can be at most eight tetrahedra in 3-space, such that every two of them
meet in a two dimensional set. We settle this conjecture affirmatively.
We get some information on families of similar nature,
consisting of eight tetrahedra. We present a joint result, showing that
there can be at most fourteen tetrahedra in 3-space, such that for every
two of them there is a plane which separates them and contains a facet of
each one of them.
Key words and phrases: Tetrahedron, neighborly family, nearly-
neighborly family, complete bipartite graph.