An Erdős–Ko–Rado theorem for cross t-intersecting families

Two families AA and BB, of k-subsets of an n-set, are cross t-intersecting   if for every choice of subsets A∈AA∈A and B∈BB∈B we have |A∩B|≥t|A∩B|≥t. We address the following conjectured cross t  -intersecting version of the Erdős–Ko–Rado theorem: For all n≥(t+1)(k−t+1)n≥(t+1)(k−t+1) the maximum value of |A||B||A||B| for two cross t  -intersecting families A,B⊂([n]k) is (n−tk−t)2. We verify this for all t≥14t≥14 except finitely many n and k for each fixed t. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a p-weight version of the problem, which comes from the product measure on the power set of an n-set.