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Updated Oct 8

Problem 1208

Statement. For $d\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\to \infty$ .

Status. Open.

Source. erdosproblems.com/1208, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1208, https://www.erdosproblems.com/1208.

References.

[CFGHUZ15] Conlon, David and Fox, Jacob and Gasarch, William and Harris, David G. and Ulrich, Douglas and Zbarsky, Samuel, Distinct volume subsets. SIAM J. Discrete Math. 29 (2015), 472-480.
[Ch13] Charalambides, Marcos, A note on distinct distance subsets. J. Geom. (2013), 439-442.
[ErGu70] Erdős, P. and Guy, R. K., Distinct distances between lattice points. Elem. Math. (1970), 121-123.
[KSS75] Komlós, J. and Sulyok, M. and Szemeredi, E., Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar. (1975), 113-121.
[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379-408.
[Th95] T. Thiele, Geometric selection problems and hypergraphs. PhD thesis, Freir Universitat Berlin (1995).

Formalization. None recorded.

Progress

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Known Results

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Linked library material

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Linked from (25)

Problem 1088 Problem 1207 Distance Problems additive_bases/croot_2026_combinatorial_large_sieve_sidon_sets_distances Theorem 1.3 (p. 3): subsets of the N by N grid with no repeated distance Theorem 1.5 (p. 4): bounded Q-distance multiplicity in [N]^2 Distance Problems distance_problems/charalambides_2013_note_distinct_distance_subsets Conjecture 2.3 (p. 2): delta(N) >= c_epsilon N^{1/2-epsilon}Proposition 1.2 (p. 1): the grid bound delta(N) << N^{1/2}(log N)^{-1/4}Proposition 2.1 (p. 1): planar distinct-distance subsets of size N^{1/3}/log N Proposition 3.1 (p. 3): distinct-distance subsets on the sphere distance_problems/conlon_2015_distinct_volume_subsets Proposition 1.1 (p. 2): every n points in R^d contain c_d n^{1/(3d-3)} (log n)^{1/3-2/(3d-3)} points with distinct distances Theorem 1.2 (p. 2): h_{a,d}(n) >= c_{a,d} n^{1/((2a-1)d)} for 2 <= a <= d+1 distance_problems/dumitrescu_2008_distinct_distances_points_general_position Theorem 3 (p. 3): planar distinct-distance subsets of size Omega(n^{0.288})distance_problems/erdos_1970_distinct_distances_between_lattice_points Inequality (1) (p. 121): the counting bound k <= n for distinct distances in the n by n grid Inequality (2) (p. 121): k < c_3 n (log n)^{-1/4} for distinct distances in the n by n grid Inequality (4) (p. 121, proved p. 122): the grid has n^{2/3-eps} points with distinct distances Inequality (7) (p. 122): k < c_7 d^{1/2} n for distinct distances among lattice points in d dimensions distance_problems/sheffer_2014_distinct_distances_open_problems_current_bounds Problem 22 (p. 11) and Theorem 6.1: the largest subset with no repeated distance, between n^(1/3)/log^(1/3) n and sqrt(n)/(log n)^(1/4)Problem 25 (p. 12): the largest subset with no repeated distance in R^d, d >= 3

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