Non-trivial intersecting uniform sub-families of hereditary families

For a family F of sets, let μ ( F ) denote the size of a smallest set in F that is not a subset of any other set in F , and for any positive integer r , let F ( r ) denote the family of r -element sets in F . We say that a family A is of Hilton–Milner (HM) type if for some A ∈ A , all sets in A ∖ { A } have a common element x ∉ A and intersect A . We show that if a hereditary family H is compressed and μ ( H ) ≥ 2 r ≥ 4 , then the HM-type family { A ∈ H ( r ) : 1 ∈ A , A ∩ [ 2 , r + 1 ] ≠ 0̸ } ∪ { [ 2 , r + 1 ] } is a largest non-trivial intersecting sub-family of H ( r ) ; this generalises a well-known result of Hilton and Milner. We demonstrate that for any r ≥ 3 and m ≥ 2 r , there exist non-compressed hereditary families H with μ ( H ) = m such that no largest non-trivial intersecting sub-family of H ( r ) is of HM type, and we suggest two conjectures about the extremal structures for arbitrary hereditary families.