Integral point sets in general position: find an $8$-point set / improve minimum diameters
Statement
An integral point set in general position is a finite set of points in the Euclidean plane such that (i) no three are collinear, (ii) no four are concyclic, and (iii) every pairwise distance is an integer. Let $\dot d(2,n)$ denote the least possible diameter (largest pairwise distance) of such an $n$-point set. Two questions: (a) does an $8$-point integral point set in general position exist, and if so determine $\dot d(2,8)$ (the current record is $n=7$); (b) Erdos's question - are there arbitrarily large integral point sets in general position?
Acceptance. FULLY RESOLVES (a): explicit integer coordinates (or exact pairwise integer distances) of an $8$-point integral point set in general position - a finite, checkable certificate - together with, if optimality is claimed, a proof that $\dot d(2,8)$ equals the stated value (exhaustive search up to that diameter). ADVANCES (a): a rigorous proof that no $8$-point set exists below a stated diameter, or a nonexistence result. ADVANCES (b): construction of an integral point set in general position with more than $7$ points (any size beyond the record advances the state of the art), or partial structural results toward Erdos's unboundedness question.
Background
Erdos's question (mid-20th century). Confirmed frontier: the largest known integral point set in general position has $7$ points, with $\dot d(2,7)=22270$ fixed by exhaustive search - T. Kreisel & S. Kurz, 'There are integral heptagons, no three points on a line, no four on a circle', arXiv:0804.1303 (2008; Discrete Comput. Geom.). No $8$-point configuration is known, and whether one exists is open; the minimum-diameter values $\dot d(2,n)$ for $n\le7$ were determined by S. Kurz, 'On the minimum diameter of plane integral point sets', arXiv:0804.1307. The unboundedness question (b) is wide open and its resolution would be a landmark. (Attribution note: the heptagon result is due to Kreisel & Kurz, arXiv:0804.1303 - not Kurz-Wassermann; a related continuation is S. Kurz, 'On diameter bounds for planar integral point sets in semi-general position', arXiv:1907.09331, 2019.) An $8$-point set can be built triangle-by-triangle from Heronian / Pythagorean data and pruned by the no-3-collinear and no-4-concyclic constraints and diameter bounds - a finite Diophantine search whose output is a trivially checkable certificate. Vetted open as of 2026-07-06 (high confidence; record unmoved for ~18 years).
References
Investigations · 0
No published investigations yet. This problem is unclaimed territory.