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Statement

Setting (pp. 129--131). $P_n$ is a set of $n$ points in the plane, and a connecting line (in the paper, simply a line) is a straight line containing at least two points of $P_n$. The $k$-th band consists of the configurations in which a largest collinear subset has exactly $n-k$ points. Write

$$M_{\max}(k)=k(n-k)+\binom k2+1,\qquad M_{\min}(k)=k(n-k)-\binom k2+1 .$$

Lemma 1 (p. 131). For all $0\le k\le n-2$, the largest number of lines determined by a configuration in the $k$-th band is $M_{\max}(k)$, and the number of lines of every configuration in the $k$-th band is at least $M_{\min}(k)$.

The paper adds (p. 132) that, with $\binom02=\binom12=0$, the lemma holds for $k=0$ and $k=1$. It attributes the lower bound to Kelly and Moser (its reference [7]); the upper bound is attained when the $k$ points off the large line are in general position (p. 130 and figure 2, p. 131).

Proof pointer

P. 132. The upper bound counts the $\binom k2$ lines among the $k$ points off the large line, the $k(n-k)$ lines joining them to the points on it, and the large line. For the lower bound, two points off the large line share at most one line through a point of it, so at least $k(n-k)-\binom k2$ of the joining lines are distinct; adding the large line gives $M_{\min}(k)$.

Read depth

Claims checked: the definitions, the statement and its range were read clause by clause on the page images of the print, and the proof on p. 132 was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input named by the paper: the lower bound of Kelly and Moser, On the number of ordinary lines determined by $n$ points, Canad. J. Math. 10 (1958), 210--219.

Source. P. Salamon and P. Erdős, The solution to a problem of Grünbaum, Canad. Math. Bull. 31 (1988), no. 2, 129--138, DOI 10.4153/CMB-1988-020-2; the edition read is named on the source card.

Bears on

• Problem 606: the lemma confines the line counts of each band to the interval from $M_{\min}(k)$ to $M_{\max}(k)$, the frame in which the paper describes the possible values band by band; on its own it does not say which values in that interval occur.