Solymosi 2010 question erdos ulam

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corollary_2_3: Every infinite rational set with no three points on a line and no four on a circle contains an infinite subset of which no algebraic curve of degree d contains more than d(d+3)/2 points.

theorem_2_1: Solymosi and de Zeeuw's main theorem: every planar point set with all pairwise distances rational has only finitely many points on an algebraic curve defined over the reals, unless the curve has a line or a circle as a component.

theorem_2_2: Solymosi and de Zeeuw's theorem that a rational set with infinitely many points on a line has at most 4 points off that line, and one with infinitely many points on a circle has at most 3 points off that circle; both numbers are attained.

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Solymosi, Jozsef and de Zeeuw, Frank, On a question of Erdős and Ulam. Discrete Comput. Geom. 43 (2010), no. 2, 393-401. DOI 10.1007/s00454-009-9179-x. The copy read for this card is the arXiv preprint arXiv:0806.3095v2 (14 January 2009); its labels and pages are the ones cited here.

A rational set is a planar point set all of whose pairwise distances are rational, and Ulam asked in 1945 whether an everywhere dense rational set exists (p. 1). Theorem 2.1 (p. 2) proves that a rational set meets a real algebraic curve in only finitely many points unless the curve has a line or a circle as a component, so no irreducible algebraic curve other than a line or a circle contains an infinite rational set; the paper presents this as proving, for algebraic curves, Erdős' conjecture that a set with a dense rational subset should be very special (abstract). Theorem 2.2 (p. 2) treats the two exceptional cases: a rational set with infinitely many points on a line has at most 4 points off that line, and one with infinitely many points on a circle has at most 3 points off that circle, which the paper says answers questions of Guy (Problem D20) and of Pach (Section 5.11 of Brass, Moser and Pach); a remark after the theorem notes that both bounds are attained, the line case by a construction of Huff and the circle case from it by inversion. Corollary 2.3 (p. 2) restates Theorem 2.1 in terms of curve-general position: every infinite rational set in general position (no 3 points on a line, no 4 on a circle) contains an infinite subset no curve of degree d meets in more than d(d+3)/2 points. The proof applies Faltings' theorem: points of a rational set are almost rational after a similarity (Lemma 3.4, p. 3), which settles curves of genus at least 2; on curves of genus 1, and of genus 0 and degree at least 4, they lift to points of an auxiliary space curve of genus at least 2; genus 0 curves of degree 2 or 3 are handled by inversion and by a hyperelliptic curve of genus 3 (Section 3.7, pp. 4-6). For Erdős problem 212 on dense rational-distance sets in the plane, the two theorems together imply (the paper does not draw this out) that a dense rational set would meet every real algebraic curve in only finitely many points; the paper does not settle the problem.

Source: https://arxiv.org/abs/0806.3095. The arXiv record names arXiv's non-exclusive distribution license (arXiv:0806.3095), every other right reserved.

Bears on. #212: Theorems 2.1 and 2.2 (p. 2) together show that a dense rational set, if one exists, has only finitely many points on each real algebraic curve, lines and circles included; this constrains a positive answer but does not settle the problem, and the paper does not draw the consequence out.

Results. Labels and pages are those of arXiv:0806.3095v2.

• Theorem 2.1 (p. 2): a rational set meets a real algebraic curve finitely often unless the curve has a line or circle component.
• Theorem 2.2 (p. 2): a rational set with infinitely many points on a line or a circle has at most 4, resp. 3, points off it.
• Corollary 2.3 (p. 2): an infinite rational set in general position has an infinite subset in curve-general position.

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