Discrete and Convex Geometry

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E0101/: Asks whether n plane points with no five on a line determine only a negligible fraction of n squared lines containing exactly four points.

E0102/: Estimates the largest number of collinear points forced when n plane points admit a constant times n squared lines with more than three points each.

E0104/: Asks whether n points in the plane lie on only a negligible fraction of n squared distinct unit circles containing three or more of them.

E0105/: Asks whether disjoint plane sets of n and n minus 3 points, the first not all collinear, admit a line meeting two points of the first and none of the second.

E0106/: Asks whether the largest total side length of interior-disjoint squares packed in the unit square equals k when there are k squared plus one squares.

E0107/: Asks whether two to the power n minus 2, plus one, points in the plane with no three collinear are always enough to force a convex n-gon.

E0173/: Asks whether every two-coloring of the plane contains a monochromatic congruent copy of every triangle, with at most one exception.

E0174/: Characterizes the finite sets of points that admit a monochromatic copy in every finite coloring of a high enough dimensional Euclidean space; answered in 2026 by OpenAI's algebraic criterion, which refutes Graham's conjecture.

E0188/: The least number of terms k such that the plane can be two-colored avoiding red points at unit distance and blue unit-spaced progressions of k terms; Erdős and Graham's question, with the step left free, has no finite answer.

E0189/: Asks whether every finite coloring of the plane has a color class containing the vertices of a rectangle of every possible area.

E0193/: Asks whether an infinite walk in the integer lattice of dimension three with steps from a finite set must contain three collinear points.

E0209/: Asks whether at least four non-parallel lines with no four meeting at a point must form a triangle whose corners each lie on only two lines.

E0210/: Asks whether the least number of lines through exactly two of n points, not all collinear, grows without bound, and how fast.

E0211/: Asks whether n points in the plane with at most n minus k on any line always determine at least a constant times k times n lines through two or more points.

E0215/: Asks whether some planar set has the property that every translated and rotated copy of it contains exactly one integer lattice point.

E0216/: Asks whether enough points in general position in the plane always contain the vertices of an empty convex k-gon, and asks for an estimate of how many are needed.

E0224/: Asks whether any two to the d plus one points in d-dimensional space must include three that form an obtuse angle.

E0352/: Asks whether some positive constant makes every planar measurable set of at least that measure contain the vertices of a triangle of area one.

E0353/: Asks whether a planar measurable set of infinite measure must contain the vertices of an isosceles trapezoid of area one, or of other prescribed shapes.

E0504/: Determines the largest angle that is guaranteed to appear among some three points of every set of n points in the plane.

E0505/: Asks whether every set of diameter one in n-dimensional space splits into at most n plus one pieces of smaller diameter.

E0506/: Determines the least number of circles determined by n points of the plane, not all on one circle or one line (Elliott's reading); known for n > 393 since Purdy and Smith's correction; a 2026 claim of every value is pending.

E0507/: Estimates the smallest area such that every set of n points in the unit disk contains three points forming a triangle of at most that area.

E0508/: Determines the fewest colors needed to color the plane so that no two points at distance one share a color.

E0526/: Characterizes which sequences of arc lengths tending to zero with infinite sum make random independent arcs cover the whole unit circle with probability one.

E0528/: The value of the limiting growth rate per step of the number of self-avoiding walks of n steps from the origin in the k-dimensional integer lattice.

E0529/: Asks whether the expected end-to-end distance of an n-step self-avoiding walk is of larger order than the square root of n in the plane, and at most of that order in every dimension at least three.

E0588/: Asks whether, for k at least four, n points in the plane with no k plus one collinear have only a negligible fraction of n squared lines through k points.

E0589/: Estimates the largest subset with no three points on a line that can always be found inside n points in the plane having no four points on a line.

E0606/: Asks which values can occur as the number of distinct lines determined by n distinct points in the plane; determined for all sufficiently large n; the small cases are open.

E0607/: Asks whether the number of distinct sets of line sizes determined by n points in the plane is at most exp(O(sqrt n)).

E0633/: Characterizes the triangles that can only be cut into a square number of congruent triangles.

E0634/: Determines all n for which some triangle can be cut into n congruent triangles.

E0651/: Asks whether the number of points in general position in k-dimensional space needed to guarantee n of them in convex position grows exponentially in n.

E0669/: Bounds how many lines can pass through at least k, or through exactly k, of n given points in the plane.

E0704/: Estimates the chromatic number of the unit distance graph in n-dimensional space, whose edges join points at distance exactly one.

E0705/: Asks whether some girth bound forces every finite unit distance graph in the plane to be 3-colorable.

E0733/: Bounds by exp(O(n^{1/2})) the number of nondecreasing sequences that can be the point counts of lines, each through at least two of n points in the plane.

E0735/: Determines when n points in the plane can be given positive weights so that every line through at least two of them has the same total weight.

E0755/: Asks whether n points in six-dimensional space span at most about one twenty-seventh of n cubed unit equilateral triangles.

E0769/: Bounds the least k beyond which the n-dimensional unit cube splits into k homothetic cubes, in particular whether it grows at least like n to the power n.

E0798/: The fewest points in the n by n grid of integers whose pairwise connecting lines cover every point of that grid.

E0827/: Estimates how many points in general position in the plane force k of them whose triples all determine circles of distinct radii.

E0831/: Estimates the least number of distinct radii among circles through three of n points in the plane, with no three collinear and no four concyclic.

E0838/: Estimates the least number of distinct convex subsets determined by n points in the plane with no three collinear, in particular whether a certain limit exists.

E0846/: Asks what follows for an infinite plane set in which every n of its points contain at least a fixed proportion with no three collinear.

E0898/: Asks whether the sum of distances from an interior point to a triangle's vertices is at least twice the sum of its distances to the three sides.

E0960/: Determines how many ordinary lines a set of n planar points with no k on a line must have to force r of the points to span only ordinary lines.

E1069/: Bounds the lines containing at least k of n plane points by n squared over k cubed for k up to root n; fails as worded at k = 1, and Szemerédi and Trotter proved it for k from 2 to root n.

E1070/: Estimates the largest guaranteed number of points, among any n points in the plane, with no two at distance one, and whether it is at least n over four.

E1071/: Asks whether a finite family of pairwise disjoint unit segments in the unit square can be maximal, so that no further unit segment can be added.

E1088/: Estimates how many points in d dimensions force n of them with all pairwise distances distinct, and whether that count is subexponential in d.

E1090/: Asks whether, for each k at least 3, some finite planar set has every two-coloring giving a line whose at least k points of the set share one color.

E1121/: Asks whether circles in the plane that no disjoint line separates can always be covered by a single circle whose radius is the sum of their radii.

E1124/: Asks whether a square and a circle of the same area can be cut into finitely many congruent pieces.

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Point configurations, lines and incidences, convex position and convex bodies, coverings, packings and geometric measure problems, together with Euclidean Ramsey theory and the colorings of the plane and of R^n descended from the Hadwiger-Nelson problem.

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