EKR Sets for Large n and r

Abstract

Let A⊂([n]r) be a compressed, intersecting family and let X⊂[n]. Let A(X)={A∈A:A∩X≠∅} and Sn,r=([n]r)({1}). Motivated by the Erdős–Ko–Rado theorem, Borg asked for which X⊂[2,n] do we have |A(X)|≤|Sn,r(X)| for all compressed, intersecting families A? We call X that satisfy this property EKR. Borg classified EKR sets X such that |X|≥r. Barber classified X, with |X|≤r, such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where A has a maximal element, we sharpen this bound to n>φ2r implies |A(X)|≤|Sn,r(X)|. We conclude by giving a generating function that speeds up computation of |A(X)| in comparison with the naïve methods.