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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.

Conjecture (Remark (4), p. 419). The authors conjecture that the lower bound of Theorem 1 is best possible for $p\ge20$, and more precisely that

$$t(p)=1+\Bigl\lfloor\frac{p(p-3)}6\Bigr\rfloor\qquad\text{for }p\ne7,11,16,19,$$

while at the four exceptional values $t(p)$ equals the lower bound of Table I and Theorem 2, that is $t(7)=6$, $t(11)=16$, $t(16)=37$ and $t(19)=52$.

The displayed equation covers every $p\ge3$ outside the four exceptions. Table I (p. 399) marks the conjectured value as proved for $p\le12$ and for $p=16$, the exceptions $7$, $11$ and $16$ included (at $p=16$ Theorem 4 gives $t(16)\le37$, an observation of this page); for the other $p\le32$ its upper and lower bounds differ by one to five. Remark (11) (pp. 421--422) notes that the conjecture, with the observation following Theorem 10, would make the pseudoline quantity exceed $t(p)$ for some $p$.

Read depth. Claims checked: the conjecture was read on the page image of the print. Nothing here is independently reviewed.

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 conjecture states the exact value of $f_3(n)$ for every $n$. The problem page records, through its claim page for Green and Tao, that $f_3(n)=\lfloor n(n-3)/6\rfloor+1$ for all large $n$, which is the conjecture for all large $n$; this page records no claim about the remaining values.