Turan Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F = {H-1,..., H-k} (k >= 1) be a family of graphs. The Turan number of the family F is the maximum number of edges in an n-vertex {H-1,..., H-k}-free graph, denoted by ex(n, F) or ex(n, {H-1, H-2,..., H-k}). The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. In this paper we determine the Turan number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Turan number of the family consisting of a cycle, a star and linear forests with k edges.