The Erdős–Ko–Rado properties of set systems defined by double partitions

Let FF be a family of subsets of a finite set VV. The star of  FFat  v∈Vv∈V is the sub-family {A∈F:v∈A}{A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r}{A∈F:|A|=r} by F(r)F(r).A double partition  PP of a finite set VV is a partition of VV into large sets that are in turn partitioned into small sets  . Given such a partition, the family F(P)F(P)induced by  PP is the family of subsets of VV whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r2r is not greater than the least cardinality of any maximal   set of F(P)F(P), then no intersecting sub-family of F(P)(r)F(P)(r) is larger than the largest star of F(P)(r)F(P)(r). We also characterise the cases when every extremal intersecting sub-family of F(P)(r)F(P)(r) is a star of F(P)(r)F(P)(r).