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Source. Theorem 3.8, p. 14, 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 deduction on p. 13 was read but not checked.

Statement

Let $S$ be a set of $n$ points in $\mathbb R^3$, no three collinear and not all coplanar. Then at least

$$\frac4{13}\binom n2$$

planes are determined by exactly three points of $S$.

Proof pointer

Page 13. Projecting the other points of $S$ from a point $p_1\in S$ onto a plane turns planes through $p_1$ and exactly two further points into ordinary lines of the projected set (Lemma 3.4, p. 10). The paper applies Csima and Sawyer's theorem on ordinary lines, which it quotes for planar sets of $n\ne7$ points not all collinear, to the $n-1$ projected points, and sums over the $n$ choices of $p_1$, each three-point plane being counted three times. The case of $n-1=7$ projected points is not discussed in the paper.

Dependencies

Lemma 3.4 of the paper and Csima and Sawyer's ordinary-line theorem.

Bears on

None of the problem pages directly. The theorem is used in the proof of Theorem 4.2.