Almost Cross-Intersecting and Almost Cross-Sperner Pairs of Families of Sets

For a set G and a family of sets F let DF (G) = {F ∈ F : F ∩ G = ∅} and SF (G) = {F ∈ F : F ⊆ G or G ⊆ F}. We say that a family is l-almost intersecting, (≤ l)-almost intersecting, l-almost Sperner, (≤ l)-almost Sperner if |DF (F )| = l, |DF (F )| ≤ l, |SF (F )| = l, |SF (F )| ≤ l (respectively) for all F ∈ F . We consider the problem of finding the largest possible family for each of the above properties. We also address the analogous generalization of cross-intersecting and cross-Sperner families. AMS subject classification: 05D05