Towards extending the Ahlswede-Khachatrian theorem to cross t-intersecting families

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, which is itself a well known extension of the t-intersecting Erd˝ os-Ko-Rado theorem. The complete intersection theorem says that the maximum size of a family of subsets of [n] = {1,. .. , n}, every pair of which intersects in at least t elements, is the size of certain trivially intersecting families proposed by Frankl. We address a cross intersecting version of their diametric theorem. Two families A and B of subsets of [n] are cross t-intersecting if for every A ∈ A and B ∈ B, A and B intersect in at least t elements. The p-weight of a k element subset A of [n] is p k (1 − p) n−k , and the weight of a family A is the sum of the weights of its sets. The weight of a pair of families is the product of the weights of the families. The maximum p-weight of a t-intersecting family depends on the value of p. Ahlswede and Khachatrian showed that for p in the range [ r t+2r−1 , r+1 t+2r+1 ], the maximum p-weight of a t-intersecting family is that of the family F t r consisting of all subsets of [n] containing at least t + r elements of the set [t + 2r]. In a previous paper we showed a cross t-intersecting version of this for large t in the case that r = 0. In this paper, we do the same in the case that r = 1. We show that for p in the range [ 1 t+1 , 2 t+3 ] the maximum p-weight of a cross t-intersecting pair of families, for t ≥ 200, is achieved when both families are F t 1. Further, we show that except at the endpoints of this range, this is, up to isomorphism, the only pair of t-intersecting families achieving this weight.