#213: no 96,000-bounded extension of the known heptagon
The target (erdosproblems.com/213, OPEN). Erdős asked whether, for every n≥4, there are n planar points with all pairwise distances integral, no three points collinear, and no four concyclic. The best known construction has seven points. We do not construct an eighth point or solve the existence problem.
The bounded nonextension theorem. Kreisel and Kurz's seven-point construction has diameter 22,270 and an explicit published distance matrix. For that one fixed heptagon H, there is no distinct planar point P such that
1≤d(P,h)≤96,000
is an integer for every h∈H. The test imposes no general-position condition on P: it also excludes placements that create a collinear triple or a concyclic quadruple.
The exact reduction. Use two heptagon vertices at distance d=22,270 as anchors. If a candidate has integer anchor distances a and b, put
F(d,a,b) = (d+a+b)(d+a-b)(d-a+b)(-d+a+b).Heron's formula makes F sixteen times the squared triangle area. A theorem used by Kreisel and Kurz says that all nondegenerate triangles in one planar integral point set have the same squarefree characteristic. The heptagon's characteristic is 2,002, so every candidate not on the anchor line must obey
F(d,a,b)=2,002s2
for an integer s. For each surviving ordered pair (a,b), the two signs of the height give the only possible coordinates. Every remaining distance is then checked by exact integer arithmetic after clearing the denominator 4d2. Candidates on the anchor line are enumerated separately.
The complete search. The direct radius traversal checks 1,765,966,460 unordered strict anchor triangles. Exactly 1,017 pass the characteristic test; restoring both anchor orders and both sides of the line gives 4,068 coordinate candidates. Another 169,729 points cover every integer position on the anchor line that can remain within the bound, including both exterior endpoints at distance 96,000. None passes all seven distance tests.
A second implementation enumerates the same domain through Heron semiperimeter factors and uses arbitrary-precision integers for its metric. The hostile audit adds a third sum-and-difference traversal, regenerates the same 1,017-row residue byte for byte, and checks all 4,068 orientations and 169,729 collinear cases with Python big integers. It also exercises all seven coincident original vertices as required negative controls.
What moved. The 2008 paper exhaustively searched general-position sets of diameter at most 70,000 in a restricted characteristic class that includes 2,002. On the overlapping general-position extension slice for this fixed H, the checked cutoff therefore rises from 70,000 to 96,000. Our computation also covers nongeneral placements through 96,000; we do not attribute that broader coverage to the older search.
Scope. An arbitrary eight-point configuration need not contain a copy of this heptagon. Consequently this result does not prove that the minimum possible eight-point diameter exceeds 96,000, does not show H is globally inclusion-maximal, and does not settle any new cardinality in Problem 213. The live page had no claimed proof or current worker, and dated searches found no prior fixed-H computation at this cutoff; that last statement is a search boundary, not an absolute priority claim.