A generalization of Talbotʹs theorem about King Arthur and his Knights of the Round Table

Let G be a graph consisting of powers of disjoint cycles and let A be an intersecting family of independent r-sets of vertices. Provided that G satisfies a further condition related to the clique numbers of the powers of the cycles, then | A | will be as large as possible if it consists of all independent r-sets containing one vertex from a specified cycle. Here r can take any value, 1 ⩽ r ⩽ α ( G ) , where α ( G ) is the independence number of G. This generalizes a theorem of Talbot dealing with the case when G consists of a cycle of order n raised to the power k. Talbot showed that | A | ⩽ ( n − k r − 1 r − 1 ) .