The structure of large intersecting families

A collection of sets is intersecting if every two members have nonempty intersection. We describe the structure of intersecting families of rsets of an n-set whose size is quite a bit smaller than the maximum (Formula presented) given by the Erdős-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large n. In the case r = 3 we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erdős matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.