PreprintCC BY 4.0arXiv
Short Proofs for the Minimum Number of Determined Circles
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≥15. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.
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BibTeX
@misc{cambie2026short,
title = {{Short Proofs for the Minimum Number of Determined Circles}},
author = {Stijn Cambie and Denis Paliy and Rafał Wrona},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18918},
eprint = {2610.18918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Short Proofs for the Minimum Number of Determined Circles
AU - Stijn Cambie
AU - Denis Paliy
AU - Rafał Wrona
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.18918
UR - https://arxiv.org/abs/2610.18918
AB - 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≥15. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.
ER -