The structure of large intersecting families

Abstract: A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$ given by the Erd\H os-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 Erdos matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.