Nontrivial independent sets of bipartite graphs and cross-intersecting families

Let G ( X , Y ) be a connected, non-complete bipartite graph with | X | ⩽ | Y | . An independent set A of G ( X , Y ) is said to be trivial if A ⊆ X or A ⊆ Y . Otherwise, A is nontrivial. By α ( X , Y ) we denote the maximum size of nontrivial independent sets of G ( X , Y ) . We prove that if the automorphism group of G ( X , Y ) is transitive and primitive on X and Y, respectively, then α ( X , Y ) = | Y | − d ( X ) + 1 , where d ( X ) is the degree of vertices in X. We also give the structures of maximum-sized nontrivial independent sets of G ( X , Y ) . Consequently, these results give the sizes and structures of maximum-sized cross-t-intersecting families of finite sets, finite vector spaces and permutations, as well as the sizes and structures of maximum-sized cross-Sperner families of finite sets and finite vector spaces.