Collinear-triple minimum in non-Desarguesian affine planes

Determine the minimum number of collinear triples, denoted Psi(A), in a generalized permutation of every finite affine plane A that is not isomorphic to AG(F,2) for any finite field F, particularly affine planes arising from non-Desarguesian projective planes.

Background

For the Desarguesian affine plane AG(F_q,2) with q a power of 2, the paper exhibits a permutation with no collinear triples, so the minimum is zero. The authors then ask for the corresponding value in affine planes not arising from finite-field vector spaces, including non-Desarguesian planes. They report computations for affine planes of order 9, where the minimum is 4 in all but one tested case and 5 for the affine plane obtained by deleting the unique translation line of the Hall plane.

References

What is $\Psi(\mathbb{A})$ for affine planes $\mathbb{A}$ other than $AG(F,2)$ for some finite field $F$, i.e., those that arise from non-Desarguesian projective planes?

— Permutations minimizing the number of collinear triples  (2501.02331 - Cooper et al., 4 Jan 2025) in Section Conclusion, Question following the discussion of affine planes and complete sets of mutually orthogonal Latin squares