Year: 2026
Type: preprint
Abstract: We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.
Source: arXiv (Cornell University)
Authors: Cambie, Stijn, Paliy, Denis, Wrona, Rafał
Institution: KU Leuven
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Fonds Wetenschappelijk Onderzoek
OpenAlex ID: W7221897616