This is a question from the textbook by Nesetril Matousek which I quote here directly but rewrite slightly:
"For a graph G let us put f(G) = α(G)·ω(G), (ω(G) denotes the maximum number of vertices of a complete subgraph of G, and α(G) is the maximum number of vertices of an independent set in G) and let us define f(n) be the minimum of f(G) over all possible G with n vertices.
For natural numbers n,k, 1 < k ≤ n/2 we define a graph Cn,k: We begin with Cn i.e., a cycle of length n, and then we connect by edges all pairs of vertices that have distance at most k in Cn. Use these graphs (with a judicious choice of k) to prove that f(n) < n for all n ≥ 7. "
I guess my question is how do I even start to think about this problem because it's not intuitive to me at all if I want to do an induction on n. So can someone give me hints on how to solve this problem.