The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem

A familyA of sets is said to be intersecting if every two sets inA intersect. An intersecting family is said to be trivial if its sets have a common element. A graph G is said to be r-EKR if at least one of the largest intersecting families of independent r-element sets of G is trivial. Let α(G) and ω(G) denote the independence number and the clique number of G, respectively. Hilton and Spencer recently showed that if G is the vertex-disjoint union of a cycle C raised to the power k and s cycles 1C, . . . , sC raised to the powers k1, . . . , ks, respectively, 1 ≤ r ≤ α(G), and