Erdos 1946 sets distances points

../

conjecture_p248: Erdős's three nested conjectures of Section 2 on points in convex position, from at least [n/2] distinct distances, through a vertex with no three vertices equidistant from it, to a point on any convex curve whose circles meet the curve at most twice.

theorem_1: Erdős's first bounds for the least number f(n) of distinct distances determined by n points in the plane, with the remark that the same method gives c_1 n^{1/k} < f(n) < c_2 n^{2/k} in k-dimensional space.

theorem_2: Erdős's bounds n^{1+c/log log n} < g(n;r) < n^{3/2} for the largest number of times one distance can occur among n points in the plane, with the remark that g(n) < n^{1+ε} seems likely.

theorem_3: Erdős's bounds on how often the largest and the smallest distance among n points in the plane can occur, with the remarks on 3n - cn^{1/2}, on Vázsonyi's 2n - 2 conjecture in space and on Borsuk's conjecture.

---

The copy read is the Renyi archive scan named below. No notice is printed (all three pages read); 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 Crossref record for DOI 10.1080/00029890.1946.11991674 (read 2026-10-02) names no license, and the publisher's page could not be read on 2026-10-02; the term is unstated.

P. Erdős: On sets of distances of $n$ points, Amer. Math. Monthly 53 (1946), 248--250 MR 7,471c; Zentralblatt 60,348.

This three-page note founds the Erdos distance problems. Theorem 1 bounds the minimum number f(n) of distinct distances determined by n points in the plane by (n - 3/4)^{1/2} - 1/2 <= f(n) <= c n/(log n)^{1/2}, the lower bound from a convex-hull vertex argument and the upper bound from the n^{1/2} by n^{1/2} integer lattice together with Landau's count of sums of two squares; the same method gives c_1 n^{1/k} < f(n) < c_2 n^{2/k} in k dimensions. Theorem 2 bounds the maximum number g(n;r) of times a single distance can repeat by n^{1+c/log log n} < g(n;r) < n^{3/2}. Theorem 3 shows the maximum distance among n planar points occurs at most n times (a bound the paper calls well known, citing the Jahresbericht DMV 43 (1934), p. 114) and the minimum distance at most 3n-6 times, the latter by planarity of the minimum-distance graph plus Euler's formula, with the remark, proof not given, that more complicated arguments give 3n - c n^{1/2}. Erdos also conjectures that n points in convex position determine at least floor(n/2) distinct distances, and the stronger statement that every convex polygon has a vertex with no three other vertices equidistant from it, which would give such a vertex floor(n/2) distinct distances to the others. The listed problems trace back here: 89 is the planar distinct-distance question at the order of Theorem 1's upper bound, 1083 the k-dimensional distinct-distance question, 90 and the planar part of 1085 the unit-distance question at the order of Theorem 2's lower bound, 223 and 1084 the maximum- and minimum-distance repetition questions (with Vazsonyi's 2n-2 conjecture in three dimensions and the link to Borsuk's problem stated explicitly), 93 the floor(n/2) convex-polygon conjecture, 982 its vertex form, and 97 a weaker form (no four vertices equidistant) of the equidistant-vertex conjecture.

Theorem 3 is also the baseline for Problem 132: the diameter always supplies one occurring distance of multiplicity at most $n$. Problem 132 asks for a second such distance and ultimately for a number tending to infinity.

Source: https://users.renyi.hu/~p_erdos/1946-03.pdf.

Read status. Claims checked: Theorems 1--3, the conjectures of Section 2 and the remarks of Sections 1, 3 and 4 were read clause by clause on the page images of pp. 248--250, and the proofs were followed; the lower-bound construction of Theorem 2, which the paper only sketches, was not re-derived. Nothing here is independently reviewed.

Bears on. #89: Theorem 1's grid upper bound $cn/(\log n)^{1/2}$ is the order the problem asks to match from below; the paper's lower bound is $(n-3/4)^{1/2}-1/2$. #1083: the remark after Theorem 1 gives $c_1n^{1/k}<f(n)<c_2n^{2/k}$ in $k$ dimensions, without proof; the problem asks whether the upper exponent is the truth. #90: Theorem 2 gives $n^{1+c/\log\log n}<g(n;r)<n^{3/2}$, the lower bound being the order the problem asks about, and the paper calls $g(n)<n^{1+\varepsilon}$ likely. #1085: Theorem 2 is the same pair of bounds for the problem's planar part. #223: Theorem 3 bounds the occurrences of the diameter in the plane by $n$, a bound the paper calls well known and cites; the paper records Vázsonyi's $2n-2$ conjecture for three dimensions. #132: by Theorem 3 the diameter is one distance occurring at most $n$ times. #1084: Theorem 3 bounds the minimum-distance pairs in the plane by $3n-6$, and the paper states $3n-cn^{1/2}$ without proof and $3n-c_1n^{1/2}$ from the triangular lattice. #93: the problem is the paper's first convex-polygon conjecture, at least $[n/2]$ distinct distances. #982: the problem is the consequence the paper draws from its equidistance-free-vertex conjecture. #97: the problem weakens that conjecture from three equidistant vertices to four.

Results. Theorem 1 (p. 248), the bounds for the fewest distinct distances, with the $k$-dimensional remark; the conjectures of Section 2 (p. 248), on points in convex position; Theorem 2 (p. 249), the bounds for the most repeated distance; Theorem 3 (p. 250), the maximum and minimum distances, with the remarks on $3n-cn^{1/2}$, Vázsonyi's conjecture and Borsuk's conjecture.

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