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Statement

Setting: bands and $M_{\max}(k)=k(n-k)+\binom k2+1$ as in Lemma 1.

Lemma 3 (p. 132). Suppose $n\le k(k+1)/2$ and $k\le n-3$. Then every integer $m$ with $M_{\max}(k)-2(n-k)\le m\le M_{\max}(k)$ is taken on, except $m=M_{\max}(k)-1$ and $m=M_{\max}(k)-3$.

The paper qualifies the lemma for the bands $k=n-2$ and $k=n-3$ (p. 133): there the moves of the proof leave the band, so the argument gives no information about those bands' structure and shows only that the values other than $M_{\max}(k)-1$ and $M_{\max}(k)-3$ in the interval occur in some band. It adds that the band $k=n-2$ has the single value $M_{\max}(n-2)=\binom n2$ and that the band $k=n-3$ has only values of the form $M_{\max}(n-3)-2j$.

The proof records the inequality $M_{\max}(k)-3\ge M_{\max}(k+1)-2(n-k)$ for all $k<n-2$ (p. 133), from which the paper concludes that the large bands overlap. The lower ends of these bands are not determined; the paper calls that information missing and apparently difficult (pp. 130--131).

Proof pointer

P. 133: the argument of Lemma 2, except that now $\binom k2\ge n-k$, so only $n-k$ points can be moved onto lines through two of the $k$ points, and the count goes down by at most $n-k$ steps of two.

Read depth

Claims checked: the statement, its hypotheses and the qualification on p. 133 were read clause by clause on the page images of the print, and the proof was followed. Nothing here is independently reviewed.

Dependencies

Lemma 2 supplies the moves used in the proof.

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 overlap of the large bands that the lemma yields is the source of the continuum of line counts leading down from $\binom n2-4$ in the paper's answer, described on the main result page.