Families intersecting on an interval

We shall be interested in the following Erdős–Ko–Rado-type question. Fix some set B⊂[n]={1,2,…,n}B⊂[n]={1,2,…,n}. How large a subfamily AA of the power set P[n]P[n] can we find such that the intersection of any two sets in AA contains a cyclic translate (modulo nn) of BB? Chung, Graham, Frankl and Shearer have proved that, in the case where B=[t]B=[t] is a block of length tt, we can do no better than taking AA to consist of all supersets of BB. We give an alternative proof of this result, which is in a certain sense more ‘direct’.