Some applications of the intersection theory to Galois geometry.

The authors state several identities and inequalities for the intersection matrix IS of a matroid S embedded in a projective space PG(n,q) . These conditions are used to prove results due to Bruck, Bose, the reviewer and Qvist on the sizes of subplanes, arcs and caps embedded in a projective space PG(n,q) . The paper is very clearly written and its results provide new methods for solving combinatorial problems of Galois geometries.