Towards extending the Ahlswede–Khachatrian theorem to cross [formula omitted]-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 Erdős–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 + 2 r − 1 , r + 1 t + 2 r + 1 ] , the maximum p -weight of a t -intersecting family is that of the family F r t consisting of all subsets of [ n ] containing at least t + r elements of the set [ t + 2 r ] . 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 1 t . 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. [ABSTRACT FROM AUTHOR]