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

For a set G and a family of sets F let D_F(G) = {F ∈ F: F ∩ G= ∅} S_F(G) ={F ∈ F: F ⊆ or G ⊆ F} We say that a family is l-almost intersecting, (≤ l)-almost intersecting, l-almost Sperner, (≤ l)-almost Sperner if {pipe}D_F(F){pipe}≤l, {pipe}S_F(F){pipe} = {pipe}S_F(F){pipe}≤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.