 
Summary: Neighborly families of boxes and bipartite coverings
Noga Alon
Dedicated to Professor Paul Erdos on the occasion of his 80th
birthday
Abstract
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.
1 Introduction
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 [3] that asserts that n noncollinear 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
[4] and many additional ones can be found in the surveys [5] and [12]. 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.
