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We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let γ: N → Q+ be any density function, i.e., γ is computable in polynomial time and satisfies γ(k) ≤ k-1 for all k ∈ N. Then γ-CLUSTER is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices that has average degree at least γ(k). For γ(k) = k - 1, this problem is the same as the well-known CLIQUE problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for γ(k)=2. We ask for the possible functions γ such that γ-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: γ CLUSTER is NP-complete if γ = 2+Ω(1/k1-ε) for some ε 0 and has a polynomial-time algorithm for γ=2+O(1/k). The algorithm also shows that for γ = 2+O(1/k1-o;(1)), γ-CLUSTER is solvable in subexponential time 2no(1).