Stability of intersecting families.

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 ≥ 2 k + 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 H M (n , k) ≔ { G ∈ [ n ] k : 1 ∈ G , G ∩ [ 2 , k + 1 ] ≠ 0̸ } ∪ { [ 2 , k + 1 ] } if n ≥ 2 k + 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 ≥ 2 k + 1. [ABSTRACT FROM AUTHOR]