Erdős-Ko-Rado with separation conditions

A family A of sets is said to be intersecting if A ∩ B = ∅ for every A,B ∈ A. For a family F of sets, let ex(F) := {A ⊆ F : A is a largest intersecting subfamily of F}. For n ≥ 0 and r ≥ 0, let [n] := {i ∈ N : i ≤ n} and ([n] r ) := {A ⊆ [n] : |A| = r}. For a sequence {di}i∈N of non-negative integers that is monotonically non-decreasing (i.e. di ≤ di+1 for all i ∈ N), let P({di}i∈N) := {{a1, . . . , ar} ⊂ N : r ∈ N, ai+1 > ai + dai for each i ∈ [r − 1]}. Let Pn := P({di}i∈N) ∩ ( [n] r ) . We determine ex(Pn) for d1 > 0 and any r, and for d1 = 0 and r ≤ 12 max{s ∈ [n] : Pn = ∅}. We particularly have that {A ∈ Pn : 1 ∈ A} ∈ ex(Pn); Holroyd, Spencer and Talbot established this for the case where d1 > 0 and di = d1 for all i ∈ N, and a part of the paper generalises a compression method that they introduced. The Erdős-Ko-Rado Theorem and the Hilton-Milner Theorem provide the solution for the case where di = 0 for all i ∈ N.