Distance Problems

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E0089/: Asks whether n distinct points in the plane always determine at least about n divided by the square root of the logarithm of n distinct distances.

E0090/: Asks whether n distinct points in the plane can have only about n pairs at distance one, up to a factor n to the power one over the log log of n.

E0091/: Asks whether, for large n, the n-point plane sets minimizing the number of distinct distances include at least two that are not similar to each other.

E0092/: Asks how many points can be equidistant from every point of an n-point plane set, and whether this maximum stays below any fixed power of n.

E0093/: Shows that n points in the plane forming a convex polygon determine at least the floor of n over 2 distinct distances.

E0094/: Bounds the sum over distances of the squared number of point pairs realizing each distance, for n points forming a convex polygon, by about n cubed.

E0095/: Bounds the sum over distances of the squared number of pairs realizing each distance, for n plane points, by n cubed times any small power of n.

E0096/: Asks whether n points in the plane forming a convex polygon have only order n pairs at distance one; false by Kruer and Kohlmeyer's Lean construction of convex point sets with unboundedly many unit distances per point.

E0097/: Asks whether every convex polygon has a vertex with no four other vertices at the same distance from it; Erdős first asked it with three, which Danzer's convex nonagon refutes.

E0098/: Asks whether the fewest distinct distances among n plane points with no three on a line and no four on a circle grows faster than n.

E0099/: Asks whether, for all sufficiently large n, an n-point plane set of minimum distance one and smallest possible diameter must contain three points forming a unit equilateral triangle.

E0100/: Asks whether n plane points whose pairwise distances are at least one and whose distinct distances differ by at least one must have diameter of order n.

E0103/: Asks whether the number of incongruent n-point plane sets of minimum distance one and least possible diameter grows without bound.

E0130/: Asks how large the chromatic and clique numbers can be for the integer-distance graph on an infinite plane set with no three collinear and no four concyclic.

E0132/: Asks whether any n points in the plane give two distances each occurring between at most n pairs, and whether the number of such distances grows.

E0135/: Asks whether a set of n points in the plane in which every four points give at least five distinct distances must determine order n squared distinct distances.

E0212/: Asks whether the plane contains a dense set of points all of whose pairwise distances are rational.

E0213/: Asks whether, for every n at least 4, there are n points in the plane with no three collinear and no four concyclic and all distances integers.

E0214/: Asks whether the complement of any planar set that avoids distance one must contain the four corners of a unit square.

E0217/: Asks for which n there are n points, no three collinear and no four concyclic, whose distances take each multiplicity up to n minus one.

E0223/: Estimates the largest number of pairs at distance one among n points of diameter one in d-dimensional space.

E0232/: Estimates the largest possible upper density of a measurable planar set containing no two points at distance one, and asks whether it is at most one quarter.

E0502/: Estimates the largest size of a set of points in n-dimensional space realizing only two distinct pairwise distances; settled to leading order: n^2/2 + O(n).

E0503/: Determines the largest size of a set of points in d-dimensional space in which every three points form an isosceles triangle.

E0604/: Asks whether any n distinct points in the plane must contain a point from which the number of distinct distances to the others is almost n.

E0605/: Asks whether n points can be placed on a sphere so that the number of pairs at one repeated distance exceeds any fixed multiple of n as n grows.

E0652/: Asks whether the least possible number of distinct distances from the k-th of n planar points, in units of root n, grows with k; Erdős's first guess, that it is unbounded already at k = 3, fails by a construction of Elekes.

E0653/: Asks whether n points in the plane can take almost n different values among the counts of distinct distances from each point to the others; yes by a Lean proof certified by Conjectures.io, unrefereed, kernel-checked by that site.

E0654/: Estimates how many distinct distances to other points some point must have, among n points in the plane with no four of them on a circle.

E0655/: Asks whether n planar points, with no circle centered at one of them holding three others, determine more than half of n distinct distances by a constant factor.

E0657/: Asks whether n planar points forming no isosceles triangle must determine a number of distinct distances that grows faster than a constant times n.

E0659/: Asks whether n planar points can have every four of them determining at least three distances while the total number of distinct distances is far below n.

E0660/: Asks whether the n vertices of a convex polyhedron in space always determine nearly half of n distinct distances.

E0661/: Asks whether two sets of n planar points can have fewer than n over the square root of the logarithm of n distinct distances between the two sets.

E0662/: Asks whether n points at mutual distance at least one have at most f(t) distances at most t, f(t) the triangular lattice's count; garbled as worded, it fails under every counting reading, and a disproof is claimed.

E0668/: Asks whether the number of incongruent n-point planar sets maximizing the number of unit distances tends to infinity, and exceeds one for every n above three.

E0670/: Asks whether n points in d-dimensional space whose pairwise distances all differ by at least one must have diameter at least (1 + o(1)) n squared.

E0754/: Estimates the largest f(n) for which some set of n points in four-dimensional space has every point equidistant from at least f(n) of the others.

E0756/: Asks whether a set of n points in the plane can determine on the order of n distinct distances each occurring for more than n pairs of points.

E0953/: The largest possible measure of a set inside a disc of radius r containing no two points at an integer distance apart.

E0956/: Determines the largest number of unit distances among n disjoint translates of a compact convex set, in particular whether it exceeds n to a power above 1.

E0957/: Asks whether the multiplicities of the smallest and largest distances among n points in the plane have product at most (9/8 + o(1)) n^2.

E0958/: Asks whether n planar points with n-1 distances of multiplicities n-1, ..., 1 must be equally spaced on a line or a circle.

E0959/: Estimates the largest possible gap between the two highest distance multiplicities determined by a set of n points in the plane.

E0982/: Asks whether n points in the plane in convex position always include a vertex with at least half of n distinct distances to the other vertices.

E1082/: Asks whether n points in the plane with no three collinear always determine at least the floor of n/2 distinct distances, even as seen from a single point.

E1083/: Estimates the least number of distinct distances determined by n points in d-dimensional space, asking whether it is nearly n to the power two over d.

E1084/: Estimates the largest number of pairs at distance exactly one among n points in d-dimensional space that are pairwise at distance at least one.

E1085/: Estimates the largest possible number of pairs at distance exactly one among n points in d-dimensional space.

E1086/: Estimates the largest number of triangles of equal area whose vertices come from a set of n points in the plane.

E1087/: Estimates the largest number of four-point subsets with two pairs at equal distance among n points in the plane, and whether it is nearly n cubed.

E1089/: Estimates the fewest points in d-dimensional space guaranteeing at least n distinct distances, and whether dividing it by d to the n minus one has a limit.

E1207/: Estimates the largest subset with no isosceles triangle guaranteed in any n points in d dimensions, in particular whether in the plane it is below a power of n.

E1208/: Estimates, for fixed dimension d, the largest number of points with all distances distinct that must lie inside every set of n points in d-dimensional space.

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Distinct, repeated and unit distances among finite point sets, diameters and nearest-neighbor conditions, and distance problems for points in convex position.

Site tags routed here: chromatic number, convex, distances, geometry, graph theory, ramsey theory.