Graphs having small number of sizes on induced k-subgraphs

Let ℓ be any positive integer, let n be a sufficiently large number, and let G be a graph on n vertices. Define, for any k, v_k(G) = |{E(H)| : H is an induced subgraph of G on k vertices}|. We show that if there exists a k, 2ℓ ≤ k ≤ n - 2ℓ, such that v_k(G) ≤ ℓ, then G has a complete or an empty subgraph on at least n - ℓ + 1 vertices and a homogeneous set of size at least n - 2ℓ + 2. These results are sharp.