A simple removal lemma for large nearly-intersecting families

A k-uniform family of subsets of [n] is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős–Ko–Rado theorem of 1961 that bounds the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs.Friedgut and Regev proved a general removal lemma, showing that when , a set family with few disjoint pairs can be made intersecting by removing few sets. Our main contribution in this paper is to provide a simple proof of a special case of this theorem, when the family has size close to . However, our theorem holds for all and provides sharp quantitative estimates.We then use this removal lemma to settle a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph K(n,k). The Erdős–Ko–Rado theorem shows . For some constant c>0 and k≤cn, we determine the sharp threshold for when this equality holds for random subgraphs of K(n,k), and provide strong bounds on the critical probability for .