Erdos 1983 combinatorial problems geometry

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conjecture_p44: Erdős's prize conjecture as the lecture states it: an infinite increasing sequence of integers whose reciprocals have divergent sum contains, for every k, an arithmetic progression of k terms.

conjecture_p46: The lecture's conjecture that if the plane is split into two classes, one class contains a triangle congruent to any given triangle, with the sole exception of a single equilateral triangle, which the strip colouring avoids.

conjecture_p53: The lecture's distinct-distance problem: Erdős thinks n points in the plane determine on the order of n/sqrt(log n) distinct distances, as the lattice attains, and records the lower bounds sqrt(n) (his own), n^(2/3) (Moser) and n^(5/7) (Chung), with a prize offer.

problem_p38: The lecture's account of ordinary lines: the number of lines through exactly two of n points not all on a line tends to infinity (Motzkin), is at least 3n/7 (Kelly and Moser), and Motzkin's conjecture of n/2 for n greater than thirteen, reported as recently proved by Hansen.

problem_p43: Erdős asks whether for every n there are n points in the plane, no three on a line and no four on a circle, with all pairwise distances integers, and reports that Harborth settled n = 5 while n = 6 was unknown.

problem_p45: The lecture's definition of a Ramsey set, its report that the unit square and every brick are Ramsey and every Ramsey set lies on a sphere, and its question whether the isosceles triangle with angles 120, 30 and 30 degrees is Ramsey.

problem_p47: Steinhaus's question as the lecture records it: whether some set S in the plane has every congruent copy containing exactly one lattice point, with Erdős's reformulation through distances sqrt(u^2+v^2) and his expectation that no such set exists.

problem_p49: The lecture's account of Esther Klein's problem: f(n) points in the plane, no three on a line, always contain a convex n-gon, with the bounds printed as 2^(n-2)+1 <= f(n) <= binomial(2n, n), Szekeres's conjecture that the lower bound is right, and f(5) = 9.

problem_p51: The lecture's variant of the convex polygon problem: n(k) is the least number such that n(k) points in the plane, no three on a line, contain a convex k-gon with no point of the set in its interior; n(4) = 5, Harborth proved n(5) = 10, and whether n(6) exists was open.

problem_p52: The lecture's account of the unit-distance problem: f_2(n), the most pairs at distance 1 among n points in the plane, satisfies f_2(n) < 2n^(3/2) and f_2(n) > n^(1+c/loglog n), with Erdős believing the lower bound right and offering a prize for f_2(n) < n^(1+e).

problem_p54: The note added on 19 August 1982: h(n) is the largest integer such that n points in the plane, no three on a line and no four on a circle, determine at least h(n) distinct distances, and Erdős asks to determine or estimate it.

question_p53: Erdős's closing question: can n points in the plane, no three on a line and no four on a circle, determine n-1 distinct distances with the i-th occurring i times; an isosceles triangle with its centre does n = 4, Pomerance's construction n = 5, and a student reportedly n = 6.

theorem_p42: The lecture's statement and proof sketch of the Anning--Erdős theorem: an infinite set of points in the plane whose pairwise distances are all integers lies on a straight line.

theorem_p47: The lecture's report of Juhász's theorem: if no two points of a set S in the plane are at distance 1, the complement of S contains a congruent copy of any four given points, settling the unit-square conjecture, while for large k the analogue for k points fails.

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Paul Erdos, Combinatorial problems in geometry. Mathematical Chronicle 12 (1983), 35-54. No notice is printed (first and last two pages read; the "(C)," on the last page is a point label in a figure); the hosting archive's site footer "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only." (https://users.renyi.hu/~p_erdos/, read 2026-10-02) speaks for the site, not the paper; the journal has no online publisher's page and no Crossref license is recorded; the term is unstated.

A transcript of Erdős's invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982; received 7 September 1982), in informal lecture style. It numbers none of its statements, so the result pages below are named by page. After Morley's theorem and the Erdős--Mordell inequality (pp. 35--36), it states the Sylvester--Gallai theorem with Kelly's proof and surveys ordinary lines (pp. 36--39); records Erdős's conjecture, which a bracketed note of 15 August 1982 says Beck proved, that n points with at most n - k on a line determine more than ckn lines (pp. 39--40), and the de Bruijn--Erdős theorem (pp. 40--41); proves the Anning--Erdős theorem that an infinite planar set with integral distances is collinear and asks for n points in general position with integral distances (pp. 41--43), with Ulam's and Besicovitch's rational-distance questions (p. 43); surveys Euclidean Ramsey theory (pp. 43--47), including the prize conjecture on progressions in sequences with divergent reciprocal sum (p. 44), Juhász's theorem (p. 47), Steinhaus's lattice-point question (pp. 47--48) and Tarski's circle-squaring problem (pp. 48--49); turns to Esther Klein's convex polygon problem and its empty-polygon variant (pp. 49--52); and ends with distances: the unit-distance problem (pp. 52--53), the distinct-distance problem with the lower bounds of Erdős, Moser and Chung (p. 53), and the closing question (pp. 53--54) whether n points, no three on a line and no four on a circle, can determine n - 1 distinct distances with the i-th occurring i times, answered for n = 4 by an isosceles triangle with its centre and for n = 5 by Pomerance's construction (the print spells the name "Pommerance" [sic]), with a reported six-point example. A note added on 19 August 1982 asks for h(n), the number of distinct distances forced by n points with no three on a line and no four on a circle. The lecture proves only the Sylvester--Gallai theorem, its corollary that n points not all on a line determine at least n lines, the Anning--Erdős theorem and small cases; every other result is reported from the literature, and the result pages say so.

Source: https://renyi.hu/~p_erdos/1983-03.pdf.

Bears on.

• Problem 217: the closing question is the problem's question; the lecture answers it yes for n = 4 (Erdős's example) and n = 5 (Pomerance's construction), reports that a Hungarian high school student did six points, and leaves larger n open (question_p53).
• Problem 98: the added note defines the problem's h(n) and asks to determine or estimate it, with no bound (problem_p54).
• Problem 89: Erdős conjectures the order n/sqrt(log n) and reports the lower bounds known in 1982 (conjecture_p53).
• Problem 90: the lecture records the lattice lower bound n^(1+c/loglog n), Erdős's belief that it is the right order, and the upper bounds known in 1982 (problem_p52).
• Problem 213: the lecture poses the problem's question and reports n = 5 settled by Harborth and n = 6 open (problem_p43; context theorem_p42).
• Problem 214: the lecture reports Juhász's theorem, which answers the problem's question yes (theorem_p47).
• Problem 3: the prize conjecture is the problem's question, stated in the affirmative (conjecture_p44).
• Problem 107: the lecture defines f(n), records Szekeres's conjecture f(n) = 2^(n-2) + 1 and reports f(5) = 9 (problem_p49).
• Problem 216: the lecture's n(k) is the problem's g(k) over points with no three on a line; it reports n(4) = 5 and n(5) = 10 and leaves n(6) open (problem_p51).
• Problem 215: the lecture records Steinhaus's question and Erdős's expectation of a negative answer, and proves nothing (problem_p47).
• Problem 173: the two-colouring conjecture is the problem's statement with the one exception named as an equilateral triangle; the lecture reports the 120-30-30 case as proved (conjecture_p46).
• Problem 174: background; the lecture reports that Ramsey sets lie on a sphere and that the unit square and bricks are Ramsey (problem_p45).
• Problem 210: the lecture reports Motzkin's theorem that the number of ordinary lines tends to infinity, the Kelly--Moser bound 3n/7 and Hansen's reported proof of n/2 for n > 13 (problem_p38).

Result pages.

• Ordinary lines (pp. 38--39): Motzkin, Kelly and Moser, and Motzkin's n/2 conjecture.
• Anning--Erdős theorem (pp. 41--43): an infinite planar set with all distances integers is collinear, with the lecture's proof.
• Integer distances in general position (p. 43).
• The prize conjecture (p. 44): divergent reciprocal sum forces arbitrarily long arithmetic progressions.
• Euclidean Ramsey sets (p. 45): definition, known results and the 120-30-30 triangle question.
• Two-colourings of the plane (p. 46): every triangle but one equilateral triangle is monochromatic.
• Juhász's theorem (p. 47): the complement of a set with no unit distance contains every four-point configuration.
• Steinhaus's lattice-point question (pp. 47--48).
• The convex polygon problem (pp. 49--50): the bounds on f(n) as printed, and f(5) = 9.
• Empty convex polygons (pp. 51--52): n(4) = 5, n(5) = 10, n(6) open.
• Unit distances (pp. 52--53).
• Distinct distances (p. 53).
• The closing question (pp. 53--54), with the four- and five-point examples.
• The added note on h(n) (p. 54).

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