Erdős Problem #100
Is the diameter of A at least Cn for some constant C > 0?
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Open problems about distances in point sets
How many distinct distances must n points determine, how often can one distance repeat, and related questions in the plane and higher dimensions. Explicit point configurations are easy to check, which makes improved constructions a good first contribution.
Erdős Problem #103
Let h(n) count the number of incongruent sets of n points in ℝ^2 which minimise the diameter subject to the constraint that d(x,y)≥ 1 for all points x≠ y. Is it true that h(n)→ ∞?
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Erdős Problem #1082
Let A⊂ ℝ^2 be a set of n points with no three on a line. Does A determine at least ⌊ n/2⌋ distinct distances?
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Erdős Problem #1083
Let d≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) - in particular, is it true that f_d(n)=n^2/d-o(1)?
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Erdős Problem #1207
Let P_d(n) be such that in any set of n points in ℝ^d there exist at least P_d(n) many points which do not contain an isosceles triangle. Estimate P_d(n) - in particular, is it true that P_2(n)<n^1-c for some constant c>0?
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Erdős Problem #212
Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
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Erdős Problem #213
Let n ≥ 4. Are there n points in ℝ^2, no three on a line and no four on a circle, such that all pairwise distances are integers?
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Erdős Problem #503
What is the size of the largest A ⊆ ℝ^n such that every three points from A determine an isosceles triangle? That is, for any three points x, y, z from A, at least two of the distances |x - y|, |y - z|, |x - z| are equal.
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Erdős Problem #653
Let x_1,…,x_n∈ ℝ^2 and let R(x_i)=\# lvert x_j-x_irvert : j≠ i, where the points are ordered such that R(x_1)≤ ⋯ ≤ R(x_n). Let g(n) be the maximum number of distinct values the R(x_i) can take. Is it true that g(n) ≥ (1-o(1))n?
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Erdős Problem #660
Let x_1, …, x_n ∈ ℝ^3 be the vertices of a convex polyhedron. Are there at least (1 - o(1)) n/2 many distinct distances between the x_i?
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Erdős Problem #757
What is the supremum of the set of admissible numbers?
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Erdős Problem #89
Erdős [Er46] asked whether every set of n distinct points in ℝ^2 determines ≫ n/√(log n) many distinct distances.
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Erdős Problem #91
Suppose A⊂ ℝ^2 has lvert Arvert=n and minimises the number of distinct distances between points in A. Prove that for large n there are at least two (and probably many) such A which are non-similar.
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Erdős Problem #959
Let A⊆ ℝ^2 be a set of size n and let d_1,…,d_k be the set of distinct distances determined by A. Let f(d) be the number of times the distance d is determined, ordered so that f(d_1)≥ f(d_2)≥ ⋯ ≥ f(d_k).
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Erdős Problem #96
If n points in ℝ^2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
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Erdős Problem #97
Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
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Erdős Problem #98
Let h(n) be such that any n points in ℝ^2, with no three on a line and no four on a circle, determine at least h(n) distinct distances. Does h(n)/n→ ∞?
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Erdős Problem #982
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
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Erdős Problem #99
For sufficiently large n, is it the case that any set of n points with minimum distance 1 that minimizes diameter must contain an equilateral triangle of side length 1?
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