Tilted Sperner families

Let AA be a family of subsets of an nn-set such that AA does not contain distinct sets AA and BB with |A∖B|=2|B∖A||A∖B|=2|B∖A|. How large can AA be? Our aim in this note is to determine the maximum size of such an AA. This answers a question of Kalai. We also give some related results and conjectures.