Martinez 2015 points defining triangles distinct circumradii

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lemma_4_1: Martínez and Roldán-Pensado's lemma that for an irreducible algebraic curve D of degree at most 6 and every integer k there is m_k = O(k^5) such that every m_k points of D in general position contain k points all of whose triples determine circles of distinct radii.

theorem_1_1: Martínez and Roldán-Pensado's theorem that if n_k is the least integer such that any n_k points in the plane with no four on a line or circle contain k points all of whose triples determine circles of distinct radii, then n_k = O(k^9).

theorem_1_2: Martínez and Roldán-Pensado's theorem that, with n_k defined for points in the plane with no four on a line or circle, n_4 is at most 9 and n_5 is at most 37.

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Martínez, L. and Roldán-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. 145 (2015), no. 1, 136--141. DOI 10.1007/s10474-014-0443-z. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1402.6276), every other right reserved. The copy read for this card is the arXiv version arXiv:1402.6276v1 (25 February 2014).

The note repairs and improves Erdős's 1978 answer to his own 1975 problem asking whether, among enough points in general position, one can always find k points such that all circles through 3 of them have distinct radii. Erdős claimed n_k at most 2 binomial(k-1,2) binomial(k-1,3) + k (so the note states on p. 1; the opening line of Section 2, p. 2, omits the factor 2), but his argument misses a case it cannot handle, a point X outside his maximal set with R(ABX) = R(CDX), and he later restated the result in 1985; Section 2 examines his argument and identifies the omission. Theorem 1.1 gives a polynomial bound: with n_k the least integer such that any n_k points in general position (no four on a line or circle, a condition the authors deliberately relax from Erdős's, since a line is a circle of infinite radius) contain k points whose triples determine circles of distinct radii, one has n_k = O(k^9); the proof, in Section 4, follows Erdős's scheme but treats the missing case using Bézout's theorem to bound the number of intersection points of two algebraic curves. Theorem 1.2 records explicit small values, n_4 at most 9 and n_5 at most 37, computed in Section 3 and used in the proof of Theorem 1.1. The note also observes that a Ramsey-theoretic route needs the existence of n_6, which is not completely trivial, and gives only an exponential tower as a bound. The paper bears on problem 827 as the corrected proof that n_k is finite, with the polynomial upper bound n_k = O(k^9); since its general position condition is weaker than Erdős's, the bound also holds for n_k defined with Erdős's condition.

Source: https://arxiv.org/abs/1402.6276.

Read status. Claims checked: Theorems 1.1 and 1.2 and Lemma 4.1 were read clause by clause on the printed pages of arXiv:1402.6276v1, and Section 2 was read for what it says Erdős's argument leaves out. The proofs (Sections 3 and 4, pp. 2-4) were read for structure only. Labels and pages on the result pages are those of that edition.

Bears on. #827: Theorem 1.1 shows that n_k exists for every k with n_k = O(k^9), and Theorem 1.2 gives n_4 at most 9 and n_5 at most 37, all under the paper's condition that no four points lie on a line or circle; every set with no three points on a line and no four on a circle meets that condition, so the bounds hold for the problem's n_k too. Lemma 4.1 is the case of point sets on one irreducible curve of degree at most 6. The paper gives no lower bound and determines n_k for no k.

Results. Theorem 1.1 (p. 1); Theorem 1.2 (p. 2); Lemma 4.1 (p. 3). Erdős's argument and the case it misses (Section 2, p. 2) are described above and on the Theorem 1.1 page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.