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Source. Theorem 4.5, p. 29, of George B. Purdy and Justin W. Smith, Lines, circles, planes and spheres, arXiv:0907.0724 (2009); Discrete Comput. Geom. 44 (2010), no. 4, 860--882, doi:10.1007/s00454-010-9270-3. Labels and pages here are those of arXiv v1 (3 July 2009), whose pagination differs from the journal's; the edition is named on the source card.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on the page image of the print; the proof was not checked.

Statement

Here $t_3^{orchard}(n)$ is the largest number of lines through exactly three points that $n$ points of the plane with no four collinear can determine (p. 25; see Theorem 4.4). The theorem as printed reads:

"Let $S$ be a set of $n$ points in $\mathbb{R}^3$, not all cospherical or coplanar, no four circular [sic] and no three collinear. If $n\geqslant 883$, then the number of spheres determined by $S$ is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$. This bound is best possible."

"Circular" stands for cocircular, the word of the hypotheses elsewhere in Section 4, as in Lemma 4.6 (p. 29). The paper asserts on pp. 26--27 that the bound is always attainable, as a consequence of Theorem 4.4. A sketch written here: $n-1$ cospherical points with no four cocircular, together with the centre $p$ of their sphere, determine that sphere and one sphere through $p$ for each triple of the $n-1$ points not coplanar with $p$; by Theorem 4.4 the $n-1$ points can be chosen so that $t_3^{orchard}(n-1)$ triples are coplanar with $p$.

Proof pointer

Pages 31--34, after Lemmas 4.6 to 4.9 on pp. 29--31. The proof is by cases on the largest number of points on a sphere or a plane. The cases of exactly $n-1$ cospherical or coplanar points follow from Lemmas 4.3 and 4.7; the cases $n-2$ and $n-3$ use inclusion and exclusion with Lemmas 4.8 and 4.9. When at most $n-4$ points lie on any plane or sphere, the proof inverts in a sphere about a point of $S$ and uses Theorem 3.11, Corollary 3.7 and Lemma 4.6; this is the case that needs $n\ge883$.

Dependencies

Theorem 3.11, Theorem 4.4, Corollary 3.7 and Lemmas 4.3 and 4.6 to 4.9 of the paper.

Bears on

None of the problem pages directly. The paper ties the theorem to the corrected circle bound in the remark on p. 8: it reports that the circle configuration was found while trying to prove a bound of $\binom{n-1}{3}$ spheres, which has a subtractive term from the orchard problem (p. 8), and that the orchard equivalence was found after a similar subtractive term was noticed for circles (p. 2).