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Source. Theorem 3.13, p. 18, 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

Let $S$ be a set of $n$ points in $\mathbb R^3$, no three collinear and at most $n-k$ coplanar, and write $m_j$ for the number of planes containing exactly $j$ points of $S$. If

$$n\ \ge\ g(k)=\Bigl(184+\frac8{25}\Bigr)k^2+4k,$$

then

$$m_3+m_4\ \ge\ k\binom{n-k}{2}-(n-k)\binom k2.$$

Proof pointer

Pages 19--20. The proof follows that of Theorem 3.11 with the degree threshold $\frac{48}{5}k$ in place of $6k$, using Lemma 3.12 (p. 18), a count of three-point planes when $r$ points lie on a plane and $s$ do not, in the first case, and Corollary 3.7 (p. 13), $m_3+m_4\ge\frac18(5m+3n)$, together with Lemma 3.10 in the second.

Dependencies

Lemmas 3.10 and 3.12 and Corollary 3.7 of the paper.

Bears on

None of the problem pages directly. The paper presents it as improving Erdős and Purdy's bound of $\frac12n^2-cn$ planes through at most four points (p. 3).