Stability of intersecting families

Abstract

The celebrated Erdős–Ko–Rado theorem (Erdős et al., 1961) states that a maximum intersecting k-uniform family on [n] must be a full star if n≥2k+1. Furthermore, Hilton and Milner (1967) showed that if an intersecting k-uniform family on [n] is not a subfamily of a full star, then its maximum size is achieved only by a family isomorphic to HM(n,k)≔{G∈[n]k:1∈G,G∩[2,k+1]≠0̸}∪{[2,k+1]} if n≥2k+1 and k≥4, and there is one more possibility if k=3. Han and Kohayakawa (2017) determined a maximum intersecting k-uniform family on [n] which is neither a subfamily of a full star nor a subfamily of the extremal families in Hilton-Milner theorem, and they asked what the next maximum intersecting k-uniform families on [n] are. Kostochka and Mubayi (2017) answered the question for large enough n. In this paper, we are going to get rid of the requirement that n is large enough in the result by Kostochka and Mubayi (2017), and answer the question of Han and Kohayakawa (2017) for all n≥2k+1.