 
Summary: COMaINAXOmCA6 (3) (1986)207219
EIGENVALUES, GEOMETRIC EXPANDERS,
SORTING IN ROUNDS, AND RAMSEY THEORY
N. ALON
Received6 June 1984
Expanding graphs are relevant to theoretical computer science in several ways. Here we
show that the points versushyperplanesincidencegraphs offinitegeometriesform highly (nonlinear)
expanding graphs with essentiallythe smallest possible number of edges.The expansion properties
of the graphs are proved using the eigenvaluesof their adjacency matrices.
These graphs enable us to improve previous results on a parallel sorting problem that arises
in structural modeling,by describing an explicit algorithm to sort n elements in k time units using
O(n~k)parallel processors, where, e.g., cq=7/4, ~q8/5, 0q=26/17 and ~q=22/15.
Our approach also yields several applications to Ramsey Theory and other extremal prob
lems in combinatorics.
1. Introduction
A graph G is called (n, ct, fl)expanding, where 0
tite graph on the sets of vertices 1 (inputs) and O (outputs), where ]l[=[Ol=n,
and every set of at least ct inputs is joined by edges to at least fl different outputs.
Expanding graphs with a small number of edges, which are the subject of
an extensive literature, are relevant to theoretical computer science in several ways.
