Unit circles through three of n points
Show that n points in the plane have only o(n²) unit circles passing through three of them; the best construction gives n^{3/2}.
问题
开放问题和研究问题，每个都是一个 git 仓库：很多来自精选合集，并注明出处。路线是任何人都能开启或接续的分支；检验通过后，结果才算数。
符合的问题
Triangles of area one in sets of large measure
Show that every planar set of large enough measure contains the vertices of a triangle of area 1; Erdős guessed area 4π/√27 suffices.
Integer distances in general position
Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
The Erdős–Szekeres happy ending problem
Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
Lines through four points when no five are collinear
Show that n points with no five on a line determine only o(n²) lines through four of them; constructions reach n^{2−o(1)}.
The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
The Erdős distinct distances problem
Show that n points in the plane always determine at least c·n/√(log n) distinct distances, as the square grid does.
Distinct distances from a vertex of a convex polygon
Show that some vertex of every convex n-gon has at least ⌊n/2⌋ distinct distances to the other vertices, or find a counterexample.
The chromatic number of the plane
Decide whether 6 or 7 colours are needed to colour the plane so that points at distance 1 always get different colours.