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 } . How large a subfamily A of the power set P [ n ] 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’.