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Statement

Here $t(p)$ is the largest number of lines through exactly three points of a $p$-point set, as defined on the Theorem 1 page.

Theorem 4 (p. 409). For every $p\ge3$,

$$t(p)\le\Bigl\lfloor\Bigl(\binom p2-\Bigl\lceil\frac{3p}7\Bigr\rceil\Bigr)\Big/3\Bigr\rfloor,$$

with $\lfloor x\rfloor$ for the print's $[x]$ and $\lceil x\rceil$ for its $]x[$, "the smallest integer $\geqslant x$".

The case $p=3$ (an observation of this page). As printed the bound fails at $p=3$: it reads $t(3)\le\lfloor(3-2)/3\rfloor=0$, while three collinear points with their line form a $(3,1)$-arrangement and Table I (p. 399) lists $t(3)=1$. The proof applies the Kelly--Moser bound, which concerns non-collinear points, to an arrangement whose points may all be collinear; for $p\ge4$ such an arrangement has $t=0$ and the bound holds, so the theorem is correct for $p\ge4$.

The paper states that Theorems 3 and 4 give all the upper bounds it knows for $t(p)$ except at $p=8,10,12,14$, which Theorems 5--8 improve, and that a better lower bound for the number of ordinary lines, for example the conjectured $t_2(p)\ge\lfloor p/2\rfloor$, would improve Theorem 4 at once (p. 409).

Read depth. Claims checked: the statement and its derivation were read on the page images of the print. The Kelly--Moser theorem is cited, not proved, in the paper and was not checked here. Nothing here is independently reviewed.

Proof pointer

P. 409. A line through exactly two of the points (an ordinary line) joins a pair that no line of the arrangement contains, so the arrangement's points have at most $e$ ordinary lines, where $e$ is the edge count of the graph $\Gamma(\mathcal A)$ of the Theorem 3 page. The paper cites Kelly and Moser (Canad. J. Math. 10 (1958), 210--219) for $t_2(p)\ge\lceil3p/7\rceil$, where $t_2(p)$ is the least number of ordinary lines of $p$ non-collinear points; then $e\ge\lceil3p/7\rceil$, and the identity $t=(\binom p2-e)/3$ gives the bound.

Source. S. A. Burr, B. Grünbaum and N. J. A. Sloane, The orchard problem, Geometriae Dedicata 2 (1974), 397--424, DOI 10.1007/BF00147569 (source card).

Bears on

• Problem 669: in the problem's notation the theorem bounds $f_3(n)$ above by $\lfloor(\binom n2-\lceil3n/7\rceil)/3\rfloor$ for $n\ge4$, which differs from the lower bound of Theorem 1 by $O(n)$. It says nothing about $F_3(n)$.