A question of Grünbaum. The Sylvester-Gallai theorem implies that if $f(m)>1$ then $f(m)\geq n$. Erdős showed that, for some constant $c>0$, all integers in $[cn^{3/2},\binom{n}{2}]$ are possible except $\binom{n}{2}-1$ and $\binom{n}{2}-3$.

Solved (for all sufficiently large $n$) completely by Erdős and Salamon [ErSa88]; the full description is too complicated to be given here.

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