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) = vertical bar{vertical bar E(H)vertical bar : H is an induced subgraph of G on k vertices}vertical bar. 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.