Dudeney no-three-in-a-line conjecture

Determine whether the maximum number of points that can be placed in an n by n grid without three collinear points is at most (pi divided by the square root of 3) times n, approximately 1.814n.

Background

The paper relates its study of collinear triples in permutation graphs to Dudeney’s classical no-3-in-a-line problem. The trivial vertical-line bound gives an upper bound of 2n points, while the sharper conjectured bound is approximately 1.814n. The authors note that even the weaker bound 2n−1 has not been established.

References

Even though it is conjectured that, in fact, no more than $\pi/\sqrt{3} n \approx 1.814 n$ points can be placed without any collinear triples (, corrected by ), not even $2n-1$ is known.

— Permutations minimizing the number of collinear triples  (2501.02331 - Cooper et al., 4 Jan 2025) in Section 1, Introduction