Stronger lower bound for generalized permutations in finite affine planes

Establish whether every finite affine plane of odd order q satisfies the lower bound Psi(A) ≥ (q−1)/2 for the minimum number of collinear triples in a generalized permutation.

Background

A previously established argument yields the lower bound Psi(A) ≥ (q−1)/4 for all finite affine planes of odd order when the vector-space proof is adapted by replacing slope classes with parallel classes. Li’s stronger bound of (q−1)/2 is known in the Desarguesian setting, but the paper leaves its validity for arbitrary finite affine planes unresolved and notes that no nontrivial upper bounds are known.

References

However, we do not know if Li's (cf.) stronger lower bound $\Psi(\mathbb{A}) \geq (q-1)/2$ holds, nor do we have any nontrivial upper bounds.

— Permutations minimizing the number of collinear triples  (2501.02331 - Cooper et al., 4 Jan 2025) in Section Conclusion, paragraph immediately following the first affine-plane question