The Cross-Intersecting Family of Certain Permutation Groups.

Two subsets X and Y of a permutation group G acting on Ω are cross-intersecting if for every x ∈ X and every y ∈ Y there exists some point α ∈ Ω such that α x = α y . Based on several observations made on the cross-independent version of Hoffman's theorem, we characterize in this paper the cross-intersecting families of certain permutation groups. Our proof uses a Cayley graph on a permutation subgroup with respect to the derangement. By carefully analyzing the cross-independent version of Hoffman's theorem, we obtain a useful theorem to consider cross-intersecting subsets of certain kinds of permutation subgroups, such as PGL (2 , q) , PSL (2 , q) and S n . [ABSTRACT FROM AUTHOR]