Maximum Hitting of a Set by Compressed Intersecting Families

For a family A and a set Z , denote {A ∈ A : A ∩ Z = ∅} by A(Z). For positive integers n and r , let Sn,r be the trivial compressed intersecting family {A ∈ ([n] r ) : 1 ∈ A}, where [n] := {1, . . . , n} and ([n] r ) := {A ⊂ [n] : |A| = r}. The following problem is considered: For r ≤ n/2, which sets Z ⊆ [n] have the property that |A(Z)| ≤ |Sn,r (Z)| for any compressed intersecting family A ⊂ ([n] r ) ? (The answer for the case 1 ∈ Z is given by the Erdős–Ko–Rado Theorem.) We give a complete answer for the case |Z | ≥ r and a partial answer for the much harder case |Z | < r . This paper is motivated by the observation that certain interesting results in extremal set theory can be proved by answering the question above for particular sets Z . Using our result for the special case when Z is the r -segment {2, . . . , r + 1}, we obtain new short proofs of two well-known Hilton–Milner theorems. At the other extreme end, by establishing that |A(Z)| ≤ |Sn,r (Z)| when Z is a final segment, we provide a new short proof of a Holroyd–Talbot extension of the Erdős-Ko-Rado Theorem.