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Claim. P. Erdős, On a problem of Grünbaum, Canad. Math. Bull. 15 (1972), no. 1, 23--25, proves that there is an absolute constant $c_1$ such that every integer $m$ with $c_1n^{3/2}<m\le\binom n2$, other than $\binom n2-1$ and $\binom n2-3$, is the number of lines determined by some $n$ points in the plane (its Theorem, p. 23). The two exceptions never occur; this is Grünbaum's observation, which the paper proves. The range is best possible up to the constant: for some $c_2>0$ and all large $n$, some $m>c_2n^{3/2}$ is not attained (p. 23, shown on p. 24 from the Kelly--Moser bound). The construction behind the Theorem places the points in general position except for a few collinear groups, whose sizes are chosen by a lemma on representing every $t<\binom n2-c_1n^{3/2}$, $t\ne1,3$, as a sum of terms $\binom{n_i}2-1$.

Covers. For all sufficiently large $n$, the values of $f(n)$ above $c_1n^{3/2}$: every integer in that range but $\binom n2-1$ and $\binom n2-3$. Not covered: the values up to $c_1n^{3/2}$, which Salamon and Erdős determine. They also show that the continuum of attained values reaches down to about $n^{3/2}$, the best constant $c=1$.

Depends on. Nothing in this wiki; the proof is the paper's own, with the Kelly--Moser theorem as its external input.

Acceptance. Refereed: the paper appeared in Canad. Math. Bull. 15 (1972), no. 1, 23--25, DOI 10.4153/CMB-1972-005-4, received 17 February 1971; the publisher's record dates the issue March 1972, the page's date. The second link is the copy on the Rényi Institute's Erdős page. Reviewed: the site's curator, T. F. Bloom, labels the problem SOLVED and credits Erdős with this range in its commentary (problem page accessed 2026-09-04). No independent proof review and no formalization are recorded.