Large induced subgraphs with equated maximum degree

For a graph G, denote by f_k (G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f₂ (G) ≥ sqrt(n) (1 - o (1)), where n is the number of vertices of G. It is conjectured that f_k (G) = Θ (sqrt(n)) for every fixed k. We prove this for k = 2, 3. While the proof for the case k = 2 is easy, already the proof for the case k = 3 is considerably more difficult. The case k = 4 remains open. A related parameter, s_k (G), denotes the maximum integer m so that there are k vertex-disjoint subgraphs of G, each with m vertices, and with the same maximum degree. We prove that for every fixed k, s_k (G) ≥ n / k - o (n). The proof relies on probabilistic arguments.