Finding cliques using few probes
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Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an n vertex graph, and need to output a clique. We show that if the input graph is drawn at random from G_n,1/2 (and hence is likely to have a clique of size roughly 2 n), then for every δ < 2 and constant ℓ, there is an α < 2 (that may depend on δ and ℓ) such that no algorithm that makes n^δ probes in ℓ rounds is likely (over the choice of the random graph) to output a clique of size larger than α n.
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