Estimate $h(n)$: distinct-radius circles forced through triples of $n$ planar points (Erdős #831)
Statement
Let $h(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$ with no three on a line and no four on a circle determine at least $h(n)$ circles of pairwise distinct radii, where each such circle passes through exactly three of the points. Estimate $h(n)$, and in particular its growth rate as $n\to\infty$.
Acceptance. FULLY RESOLVES: determine the asymptotic order of $h(n)$ (matching lower and upper bounds up to constant or logarithmic factors), with proof — machine-checkable preferred, else a complete written proof. ADVANCES: prove a nontrivial lower bound $h(n)\gg g(n)$ for an explicit increasing function $g$, strictly better than any bound stated in the background, with proof; OR give a construction (with proof, or a reproducible and certified computation) of $n$-point configurations achieving few distinct triple-circle radii, establishing a nontrivial upper bound on $h(n)$; OR compute exact/extremal values of $h(n)$ for small $n$ with a reproducible search and a certificate. Deliver the proof, or the construction/search code plus the configurations and certificate.
Background
Posed by Erdős [Er75h] and [Er92e]; listed as open on erdosproblems.com/831 (fetched 2026-07-13, status 'open', tagged 'geometry'). No prize. Any such configuration of $n$ points determines $\binom{n}{3}$ triples, and since no four points are concyclic these give $\binom{n}{3}$ distinct circles; the question is how many of them are forced to have pairwise distinct radii — a circle-radius analogue of Erdős's distinct-distances problem. The cited record states no explicit bounds, so even the correct order of magnitude of $h(n)$ is open. The site relates the problem to Erdős #104 and Erdős #506 (erdosproblems.com/104, erdosproblems.com/506) on distinct distances and repeated configurations, and it is the extremal-counting counterpart of the venue's Ramsey-type sibling Erdős #827 (fewest points forcing a $k$-subset whose triples give all-distinct radii). Attacker's tool: computer search for point configurations that minimise the number of distinct triple-circle radii (yielding upper-bound constructions for $h(n)$), lower-bound arguments via incidence and distinct-distances machinery, and numerical probing of exact values for small $n$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #831 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.