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Source. Lemma 4.1, p. 3, of L. Martínez and E. Roldán-Pensado, Points defining triangles with distinct circumradii, Acta Math. Hungar. 145 (2015), no. 1, 136-141, doi:10.1007/s10474-014-0443-z; read in arXiv:1402.6276v1 (25 February 2014), the edition named on the source card. Pages are those of that edition.

Read depth. Claims checked: the statement was read clause by clause on the printed page, and the proof (pp. 3-4) was read for structure only. Nothing here is independently reviewed.

Statement

General position is the paper's: no four points on a line or circle.

Lemma 4.1 (p. 3, quoted). "Let $\mathcal{D}$ be an irreducible algebraic curve of degree at most $6$. Then for every integer $k$ there exists an integer $m_k = O(k^5)$ such that the following holds: every set $\mathcal{F} \subset \mathcal{D}$ with $m_k$ points in general position contains a subset $\mathcal{G}$ with $k$ points such that all its triples determine circles of distinct radii."

The paper calls the lemma a particular case of its main theorem (p. 3). The degree $6$ matches the curves $\mathcal C(AB,CD)=\{X: R(ABX)=R(CDX)\}$ of Theorem 1.1, to which the lemma is applied.

Proof pointer

Pages 3-4. Take a maximal subset $\mathcal G$ of the $m$ points with all triples of distinct circumradii, $l=|\mathcal G|$, and assume $l\ge5$ by Theorem 1.2. Points on circles of the $\binom l3$ radii through two points of $\mathcal G$ number at most $2\binom l2\binom l3$. For two distinct pairs $\{A,B\},\{C,D\}$ of $\mathcal G$, Bézout's theorem gives that either $\mathcal D$ is an irreducible component of $\mathcal C(AB,CD)$ or the two curves meet in at most $36$ points; the first case would force $\mathcal G=\{A,B\}\cup\{C,D\}$, against $l\ge5$. Hence $m-l\le2\binom l2\binom l3+36\binom{\binom l2}{2}$, which gives $m_k=O(k^5)$.

Dependencies

Theorem 1.2 of the same paper and Bézout's theorem.

Bears on

• Problem 827: the lemma is the special case of the problem for point sets lying on one irreducible algebraic curve of degree at most $6$, under the paper's general position condition. On its own it bounds $n_k$ for no general point set; it is the step that handles the case Erdős's 1978 argument leaves out.