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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 3 (p. 409). For every $p\ge3$,

$$t(p)\le\Bigl\lfloor\frac p3\Bigl\lfloor\frac{p-1}2\Bigr\rfloor\Bigr\rfloor,$$

with $\lfloor x\rfloor$ for the print's $[x]$, "the greatest integer $\leqslant x$".

The graph $\Gamma(\mathcal A)$ (p. 408), used again in Theorems 4--8. For a $(p,t)$-arrangement $\mathcal A$, $\Gamma(\mathcal A)$ has the $p$ points as nodes, two of them adjacent exactly when no line of $\mathcal A$ contains both. A point on $k$ lines of $\mathcal A$ has valence $p-1-2k$, so every valence has the parity opposite to $p$, and $\Gamma(\mathcal A)$ has $e=\binom p2-3t$ edges, that is $t=(\binom p2-e)/3$ (the paper's $(*)$). Remark (6) (p. 420) notes that Coxeter calls $\Gamma(\mathcal A)$ the Menger graph of $\mathcal A$.

Read depth. Claims checked: the statement and its derivation were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Proof pointer

Pp. 408--409. From $(*)$ and $e\ge0$, $t\le\lfloor\binom p2/3\rfloor$ for all $p$. For even $p$ every node has odd, hence positive, valence, so $e\ge p/2$ and $t\le\lfloor p(p-2)/6\rfloor$. The theorem writes the two cases as one formula. The argument is purely combinatorial, which Theorem 9 uses to carry it to pseudolines.

Remark (7) (pp. 420--421) calls a purely combinatorial generalization of Theorem 3 the question of $T(p)$, the largest number of triples on $p$ symbols with no pair in two triples. It records Kirkman's and Schönheim's values of $T(p)$ (listed in Table I, p. 399) and says that Theorem 3 strengthens to $t(p)\le T(p)$, with a pseudoline analogue of $t(p)$ between them.

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 $\lfloor\frac n3\lfloor\frac{n-1}2\rfloor\rfloor$, which is the pair count for lines through exactly three points with a parity correction for even $n$. It says nothing about $F_3(n)$, which also counts lines through more than three points.