One of the seminal results in extremal combinatorics, due to Erdo&huml;s, Ko and Rado, states that if [special characters omitted] is an intersecting family of r-subsets of an n-element set, i.e. for any A,B ∈ [special characters omitted], A[special characters omitted]B ≠ [special characters omitted], then |[special characters omitted]| ≤ [special characters omitted] if r ≤ n/2. Furthermore, when r < n/2, the only structure which attains this extremal number is that of a star. A major part of this dissertation considers extensions of the Erdo&huml;s–Ko–Rado theorem motivated by a graph-theoretic generalization due to Holroyd, Spencer and Talbot. A conjecture of Holroyd and Talbot is proved for a large class of graphs, namely chordal graphs which contain at least one isolated vertex. A stronger result is also shown to exist for a special class of chordal graphs obtained by blowing up edges of a path into complete graphs.

Next, a well-known generalization of the EKR theorem due to Frankl is considered. For some k ≥ 2, let [special characters omitted] be a k-wise intersecting family of r-subsets of an n-element set, i.e. for any F ₁, …, F_k ∈ [special characters omitted] F_i ≠ [special characters omitted]. If r ≤ [special characters omitted] then |[special characters omitted]| ≤ [special characters omitted]. A stability version of this theorem is proved using an analog of Katona's circle method. A graph-theoretic generalization of Frankl's theorem analogous to a theorem of Bollobás and Leader is also formulated and proved.

A collection of families [special characters omitted] is called cross-intersecting if for any i, j ∈ [k] with i ≠ j, A ∈ [special characters omitted] and B ∈ [special characters omitted] implies A [special characters omitted] B ≠ [special characters omitted]. Hilton proved a best possible upper bound on the sum of the cardinalities of uniform cross-intersecting subfamilies. In this thesis, extensions of Hilton's theorem are formulated and proved for chordal graphs and cycles.

One of the motivations in formulating these graph-theoretic generalizations for EKR theorems is a long-standing conjecture of Chvátal for hereditary set systems. A set system [special characters omitted] is said to be hereditary if for any F ∈ [special characters omitted], if G ⊆ F, then G ∈ [special characters omitted]. Chvátal's conjecture states that the set of maximum-sized intersecting subfamilies of a hereditary set system contains a star. It can be observed that the family of all independent sets in a graph is hereditary. A different class of hereditary vertex families in a graph is studied, namely the family of all cycle-free vertex subsets of a graph. Finally, a powerful tool of Erdo&huml;s and Rado is used to prove Chvátal’s conjecture for hereditary families with small rank.