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Statement

Here $\tilde t(n)$ is the largest number of vertices lying on exactly three pseudolines in an arrangement of $n$ pseudolines, as defined on the Theorem 9 page.

Theorem 10 (p. 417). For all $k\ge3$,

$$\tilde t(2k)\ge\binom k2+\tilde t(k).$$

Consequences drawn in the paper (p. 417). With $\tau(n)=1+\lfloor n(n-3)/6\rfloor$, the bound of Theorem 1, one has $\tau(2k)=\binom k2+\tau(k)$, so $\tilde t(k)>\tau(k)$ implies $\tilde t(2k)>\tau(2k)$. Since Theorem 2 gives $t(k)>\tau(k)$ for $k=7,11,16,19$, the paper concludes $\tilde t(2^jk)>\tau(2^jk)$ for those $k$ and all $j\ge0$. In particular $\tilde t(14)\ge27$, against the line value $t(14)\ge26$; Table I (p. 399) lists the resulting bounds $\tilde t(22)\ge71$, $\tilde t(28)\ge118$ and $\tilde t(32)\ge157$.

Open questions (Remark (11), pp. 421--422). The authors could not prove $\tilde t(p)\ne t(p)$ for any $p$, though their conjecture in Remark (4) together with the observation following Theorem 10 would give $\tilde t(p)>t(p)$ for some $p$. They also ask whether $\tilde t(p)-(1+\lfloor p(p-3)/6\rfloor)$ is bounded, possibly by $2$, the most their examples give, or unbounded along some sequence of $p$. Remark (10) (p. 421) defines a second pseudoline variant, from $p$ chosen vertices and pseudolines through three of them, states the same doubling bound for it, and conjectures that the two variants agree for all $p$.

Read depth. Claims checked: the statement and the consequences were read on the page images of the print. The construction is described in words and by Figure 20 for $k=7$ and was not checked in general here. Nothing here is independently reviewed.

Proof pointer

Pp. 416--417. The paper starts from an arrangement $\mathcal A_1(2k)$ of $2k$ lines, the $k$ edge lines of a regular $k$-gon and its $k$ lines of symmetry, which has $\binom k2$ triple points and one point on $k$ of its lines (cited from Grünbaum's 1971 paper on arrangements of hyperplanes, p. 75). It removes a small disc around that $k$-fold point and reroutes the $k$ lines through it inside the disc as an arrangement of $k$ pseudolines with $\tilde t(k)$ triple points. Figure 20 (p. 416) draws $k=7$.

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

No Erdős problem directly: the problems the paper bears on concern points and straight lines in the plane, and the theorem gives nothing for straight lines.