Some set intersection theorems of extremal type

For a family F of sets, let ex(F) := {A : A is an extremal intersecting sub-family of F}. The Erd®s-Ko-Rado (EKR) Theorem states that {A ∈ ( [n] r ) : 1 ∈ A} ∈ ex( ( [n] r ) ) if r ≤ n/2. The Hilton-Milner (HM) Theorem states that if r ≤ n/2 and A is a nontrivial intersecting sub-family of ( [n] r ) then |A| ≤ |{A ∈ ( [n] r ) : 1 ∈ A,A ∩ [2, r + 1] 6= ∅} ∪ {[2, r + 1]}|; hence {{A ∈ ( [n] r ) : j ∈ A} : j ∈ [n]} = ex( ( [n] r ) ) if r < n/2. Thus we say that a family F is (strictly) EKR if ex(F) contains (only) trivial intersecting families. We obtain a partial solution to the following problem: for r ≤ n/2, which sets Z ⊆ [n] have the property that |{A ∈ A : A∩Z 6= ∅}| ≤ |{A ∈ ( [n] r ) : 1 ∈ A,A∩Z 6= ∅}| for all compressed intersecting sub-families of ( [n] r ) ? Using the idea of this problem, we generalise the HM Theorem to a setting of compressed hereditary families. For a set X := {x1, ..., x|X|}, we de ne the family SX,k of signed sets by SX,k := {{(x1, a1), ..., (x|X|, a|X|)} : a1, ..., a|X| ∈ [k]} and the sub-family S∗ X,k by S∗ X,k := {{(x1, a1), ..., (x|X|, a|X|)} : {a1, ..., a|X|} ∈ ( [k] |X| ) }. For a family F , let SF ,k := ⋃ F∈F SF,k, S∗ F ,k := ⋃