Families intersecting on an interval

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