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Source. Problem 36, p. 102, of P. Erdős, Research problems, Period. Math. Hungar. 15 (1984), no. 1, 101--103, doi:10.1007/BF02109375. The edition read is named on the source card.

Statement

Definition (p. 102). $\alpha_1<\alpha_2<\cdots$ are the integers $\alpha$ for which some set $X_n$ of $n$ points in the plane determines exactly $\alpha$ distinct lines.

What the note says (p. 102). Several results on the possible values are known, citing Erdős's 1972 paper On a problem of Grünbaum; for example $\alpha_1=1$ and $\alpha_2=n$. As far as Erdős knows, the number of possible values of the $\alpha_i$ has not been determined. He adds that much less seems to be known about the possible numbers of ordinary lines, or of lines containing exactly $r$ of the points.

Proof pointer

None; the note states the known values without argument.

Read depth

Claims checked: the paragraph was read clause by clause on the page image of p. 102. Nothing here is independently reviewed.

Dependencies

The known values are cited to P. Erdős, On a problem of Grünbaum, Canad. Math. Bull. 15 (1972), 23--25, which the library does not hold.

Bears on

• Problem 606: the problem asks for the possible values of the number of distinct lines determined by $n$ points in the plane, which are the note's $\alpha_i$. The note records only $\alpha_1=1$, $\alpha_2=n$ and that the count of possible values is open.