Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to estimate $f(m)$. When $m$ is near the ends of the interval, it is not too difficult. The case of interest occurs when $m = q^a$, and $1 < a < 2$. Are estimates of $f(m)$ known?
