Extremal Problems Related To Erd(?)s-Ko-Rado Theorem

This thesis investigates the sizes and structures of independent sets in direct product of vertex-transitive graphs, intersecting antichains in linear lattices, and analogs of Erdos-Ko-Rado Theorem in labeled sets.The thesis consists of four chapters. The first contributes to the contents of Erdos-Ko-Rado Theorem and several classic theorems related to it. Then the concept of sym-metric systems is introduced, and the equivalence between symmetric systems and vertex transitive graphs is established. Hence, an intersecting family in symmetric systems cor-responds to an independent set of a vertex transitive graph. In the second chapter, we prove a(Circ(r,n)×H)=max{r|H|,na(H)}, for every vertex transitive graph H and the circular graph Circ (r,n), and identify the structure of maximum independent sets in Circ(r,n)×H. As consequences, we determine a(GxH) for G being Kneser graphs and the graphs defined by permutations and partial permutations. respectively. The structures of maximum independent sets in these direct products are also identified. The third chapter investigates t-intersecting antichains in linear lattices. We establish a homoinorphism of finite linear lattices onto the Boolean lattices via a group acting on linear lattices. By us-ing this homomorphism we prove that the t-intersecting antichains in finite linear lattices satisfy an LYM-type inequality. The fourth chapter considers analogs of Erdos-Ko-Rado Theorem in labeled sets. In this chapter we give the sizes and structures of intersecting families of labeled r-sets by using the shift operator in an inductive argument.