Erdos 1978 set theoretic

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assertion_p122: Records Erdős's assertion that a plane set of infinite planar measure contains, for every a > 0, three points spanning a triangle of area a, which may be taken isosceles or right-angled, while some set of infinite planar measure contains no equilateral triangle of unit area; no proof is printed.

conjecture_p123: Records Erdős's long-standing conjecture that for every infinite set A on the line some set of positive measure contains no set similar to A, with the finite case, due substantially to Steinhaus, and the follow-up question on the largest measure of such a set in [0,1].

question_p122: Records Erdős's question whether some absolute constant C makes every plane set of measure greater than C contain the vertices of a triangle of area 1, with his example of the disc of radius 23^{-3/4}, of area 4pi3^{-3/2}, which contains no such triangle and which he suggests may give the right C.

theorem_1: States Erdős's Theorem 1 that a subset S of k-dimensional Euclidean space with |S| = m >= aleph_0 has a subset of cardinality m in which all distances between points are distinct, proved without the continuum hypothesis.

theorem_2: States Erdős's Theorem 2 that when the continuum exceeds aleph_1, in every decomposition of the real line into countably many sets some set determines a distance twice, with the Erdős-Hajnal lemma on colorings of K(A,B) used to prove it.

theorem_p133: States Erdős's result that when the continuum exceeds aleph_1 and each of countably many sets of reals has all its pairwise sums distinct, the complement of their union contains a translate of the rational span of aleph_1 rationally independent reals.

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P. Erdős: Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79) no. 2, 113--138 MR 80g:04005; Zentralblatt 418.04002; doi:10.2307/44151159.

This topical survey collects problems where geometry, number theory and set theory meet, with detailed proofs supplied where published ones are hard to find. Theorem 1 (p. 114) states that any subset S of k-dimensional Euclidean space with |S| = m >= aleph_0 has a subset S_1 of the same cardinality all of whose distances are distinct; the proof, redone here without the continuum hypothesis and repairing a gap pointed out by Bollobás and others, inducts on |S| and on the dimension, takes n = cf(m) hyperplanes or hyperspheres of least dimension that together meet S in m points, uses the Dushnik-Miller partition relation n -> (n,k)^2 to keep n of them pairwise non-orthogonal, and builds the subset by transfinite induction. The survey then contrasts this with hard finite analogs (the conjectures on f_1(n) and g_1(n), Croft's n_3 = 9, and the Larman-Rogers-Seidel bound max|S_k^{(2)}| = k^2/2 + O(k) for sets with at most two distances), records the Erdős-Kakutani equivalence of c = aleph_1 with the real line being a countable union of Hamel bases, and notes Davies' result for the plane and (added in proof) Kunen's for all k that, under c = aleph_1, k-space is a union of countably many sets with all distances distinct. On the measure-theoretic side it notes, leaving the proof via the Lebesgue density theorem as an exercise, that a plane set of infinite measure contains the vertices of a triangle of any prescribed area, and that some set of infinite measure has no unit-area equilateral triangle, and states the Erdős similarity conjecture that every infinite A on the line is avoided up to similarity by some set of positive measure. Problem 352 is the question raised on pages 122-123: is there an absolute constant C such that every plane set of measure greater than C contains the vertices of a triangle of area 1, where Erdős notes the disc |z| < 23^{-3/4} of area 4pi3^{-3/2} contains no such triangle and may give the correct C. Section 2 proves Theorem 2 (p. 127): if c > aleph_1, every decomposition of the line into countably many sets has a set with a repeated distance.

Source: https://users.renyi.hu/~p_erdos/1978-40.pdf. No notice is printed; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the publisher's page could not be read on 2026-10-02 (Project Euclid returned only a bot-detection page), and the Crossref record for doi:10.2307/44151159 records no license; the term is unstated.

Bears on.

• #352: the problem is the question of pages 122-123; the paper gives the disc example, showing any such constant is at least 4pi*3^{-3/2}, and does not answer it (the question).
• #120: the problem is the similarity conjecture of p. 123, stated as open (the conjecture).
• #353: the paper asserts without proof (p. 122) that a plane set of infinite measure contains isosceles and right-angled triangles of every area; these are the problem's two triangle variants, and the paper says nothing on its other parts (the assertion).
• #1127: the paper states the problem as Erdős's conjecture under c = aleph_1 and reports Davies's proof for the plane and Kunen's for all k (p. 121); Theorem 2 shows that the line has no such decomposition when c > aleph_1 (Theorem 2).
• #465 and #466: the paper states Erdős's conjectures that N(x,delta) = o(x) for every 0 < delta < 1/2 (the first question of #465; the paper does not ask the second) and that N(x,delta_0) tends to infinity for some delta_0 > 0 (#466). It reports, without proof, that Sárközy proved the first with N(x,delta) < (4*10^4/delta^3) x/log log x, that Graham proved the second with N(x,1/10) > (log x)/10, and Sárközy's lower bounds N(x,1/10) > x^c and N(x,delta) > x^{1/2-epsilon} for delta < delta(epsilon) (pp. 123-124); no page here.
• #214: the paper reports, without proof, Juhász's theorem that the complement of a plane set with no two points at distance one contains a congruent copy of every four-point set (p. 126); no page here.

Results. Labels and pages are those of the journal print.

• Theorem 1 (p. 114): every infinite subset of E_k has a subset of the same cardinality with all distances distinct, proved without the continuum hypothesis.
• Triangles in sets of infinite measure (p. 122): every area occurs, also for isosceles or right-angled triangles; some such set has no unit-area equilateral triangle. Asserted without proof.
• The triangle question (pp. 122-123): does planar measure greater than an absolute C force a triangle of area 1? The disc of area 4pi*3^{-3/2} has none.
• The similarity conjecture (p. 123): every infinite set on the line is avoided, up to similarity, by some set of positive measure; the finite case is Steinhaus's.
• Theorem 2 (p. 127): if c > aleph_1, a countable decomposition of the line has a set with a repeated distance; with the Hajnal-Erdős lemma (p. 128) on countable colorings of K(A,B), |A| = aleph_2, |B| = aleph_1.
• The complement theorem (pp. 133-135): if c > aleph_1, the complement of countably many sets of reals with distinct pairwise sums contains a translate of an aleph_1-dimensional rational subspace.

The survey's other statements are reported results of others (Larman, Rogers and Seidel on two-distance sets, p. 120; Erdős and Kakutani on Hamel bases, pp. 120-121; Euclidean Ramsey theory, pp. 125-126; Section 3, pp. 136-138) or problems with no page here.

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