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Statement

Setting (p. 415, Section 4). An arrangement of $n$ pseudolines is a family of $n$ simple closed curves in the real projective plane, every two of which cross at exactly one point, together with their crossing points (its vertices); $v_k(\mathcal C)$ is the number of vertices on exactly $k$ of the curves, and $\tilde t(n)$ is the largest $v_3(\mathcal C)$ over arrangements $\mathcal C$ of $n$ pseudolines. By projective duality the orchard number $t(p)$ of the Theorem 1 page is the largest number of triple points of an arrangement of $p$ lines, and since lines are pseudolines, $\tilde t(p)\ge t(p)$ for all $p$ (p. 416).

Theorem 9 (p. 416, quoted). "All the results of Section 3 are valid for arrangements of pseudolines."

Section 3 holds Theorems 3--8, so the theorem asserts the bounds of Theorem 3 and Theorem 4 for $\tilde t(p)$, and the non-existence statements of Theorems 5--8 for pseudolines, giving $\tilde t(8)=7$, $\tilde t(10)=12$, $\tilde t(12)=19$ and $\tilde t(14)\le27$. Table I (p. 399) gives one column of upper bounds for both $t(p)$ and $\tilde t(p)$. The failure of Theorem 4's printed bound at $p=3$, recorded on its page, carries over, since three concurrent lines form a pseudoline arrangement with one triple point.

Read depth. Claims checked: the definitions and the statement were read on the page images of the print. The proof is a short argument about the earlier proofs and was not checked against them case by case; the Kelly--Rottenberg theorem is cited, not proved. Nothing here is independently reviewed.

Proof pointer

P. 416. Theorem 3 is purely combinatorial. For Theorem 4 the paper replaces the Kelly--Moser bound on ordinary lines by the theorem of Kelly and Rottenberg (Pacific J. Math. 40 (1972), 617--622) that every arrangement of $p$ pseudolines has $v_2(\mathcal C)\ge\lceil3p/7\rceil$. For Theorems 5--8 it states that their proofs used only separation and order properties, never straightness, so the dual proofs apply to pseudolines.

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.