Erdős Problem #506
DECIDABLEdatabase updated 2025-08-31
Still open, but reduced to a finite computation: checking enough cases would settle it either way.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The number of circles determined by a set of points `P` in the plane: the number of distinct circles (spheres) passing through at least three of the points of `P`. (Three distinct points lie on a common circle precisely when they are not collinear, so this counts exactly the circumcircles of the non-collinear triples of `P`.)
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/506.Prizenone
Topicsgeometry
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/506
Attempts
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Under Elliott’s intended “not all collinear or concyclic” convention, \(f(7)=11\) is proved with an exact integer construction and a from-scratch lower bound; also \(12\leq f(8)\leq18\), with the exac
PARTIAL