The maximum density of a no-three-in-line set on the checkerboard is at most the middle real root α of the cubic equation 401α³ - 1744α² + 2240α - 768 = 0.
Flammenkamp, Progress in the no-three-in-line problem, J
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No-three-in-line sets on the checkerboard grid
The maximum density of a no-three-in-line set on the checkerboard is at most the middle real root α of the cubic equation 401α³ - 1744α² + 2240α - 768 = 0.