Summary: Neighborly families of boxes and bipartite coverings
Dedicated to Professor Paul Erdos on the occasion of his 80th
A bipartite covering of order k of the complete graph Kn on n vertices is a collection
of complete bipartite graphs so that every edge of Kn lies in at least 1 and at most
k of them. It is shown that the minimum possible number of subgraphs in such a
collection is (kn1/k). This extends a result of Graham and Pollak, answers a question
of Felzenbaum and Perles, and has some geometric consequences. The proofs combine
combinatorial techniques with some simple linear algebraic tools.
Paul Erdos taught us that various extremal problems in Combinatorial Geometry are best
studied by formulating them as problems in Graph Theory. The celebrated Erdos de Bruijn
theorem  that asserts that n non-collinear points in the plane determine at least n distinct
lines is one of the early examples of this phenomenon. An even earlier example appears in
 and many additional ones can be found in the surveys  and . In the present note
we consider another example of an extremal geometric problem which is closely related to a
graph theoretic one. Following the Erdos tradition we study the graph theoretic problem in
order to deduce the geometric consequences.