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Statement

Setting: bands and $M_{\max}(k)=k(n-k)+\binom k2+1$ as in Lemma 1; $[x]$ is the integer part of $x$.

Lemma 4 (p. 133). For $n$ sufficiently large, every configuration of $n$ points in a band with $k>[\sqrt{n+2}]$ determines more than $M_{\max}([\sqrt{n+2}]-1)$ lines.

No explicit threshold for $n$ is given.

Proof pointer

P. 133. The Kelly--Moser lower bounds $M_{\min}(k)$ of Lemma 1 increase up to $k=[(n+0.5)/3]$ and, for $k>[\sqrt{n+2}]$, exceed $M_{\max}([\sqrt{n+2}]-1)$, so only bands with $k>n/3$ remain. For those, the theorem of Beck that the paper cites (a configuration with $k\ge x$ determines more than $c\,x(n-x)$ lines, $c$ absolute; p. 131) gives at least $c(n/3)(2n/3)$ lines, which exceeds $M_{\max}([\sqrt{n+2}]-1)$, of order $n^{3/2}$, once $n$ is large.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print, and the proof on p. 133 was followed. Beck's theorem is cited, not proved, in the paper and was not read. Nothing here is independently reviewed.

Dependencies

Lemma 1 for the lower bound $M_{\min}(k)$. External input named by the paper: J. Beck, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry, Combinatorica 3 (1983), 281--297, with Szemerédi and Trotter, Combinatorica 3 (1983), 381--392.

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 is the reason the bands beyond $k=[\sqrt{n+2}]$ contribute nothing below the continuum in the paper's answer for large $n$, described on the main result page. The lemma gives no threshold for $n$, and that answer is stated only for $n\ge n^*$, with $n^*$ unknown (p. 137).