Permutations minimizing the number of collinear triples
Joshua Cooper, Jack Hyatt
TL;DR
The paper resolves the problem of minimizing collinear triples in graphs of permutations over $\mathbb{F}_q$ by embedding the permutation graph in the projective plane $PG(2,q)$ and showing the minimizing configuration must lie on a nondegenerate irreducible conic with an external point. This structure forces the permutation to be an extended linear fractional transformation (ELFTP) $\sigma(x)=\frac{\alpha x+\beta}{x+\gamma}$, and the authors prove that for odd prime powers $q$, any permutation achieving the bound $\frac{q-1}{2}$ collinear triples is an ELFTP; among these, the lexicographically least one with $\sigma(0)=0$, $\sigma(1)=1$, and $\sigma(2)=2$ is unique and explicitly determined. The work connects these algebraic-geometric characterizations to broader themes such as MOLs and generalized permutations in finite affine planes, and it raises open problems about $\Psi(\mathbb{A})$ for non-Desarguesian planes and composite moduli, highlighting implications for finite geometry and combinatorial design. Overall, the results provide a precise, structurally rich description of extremal permutations in finite affine geometries and offer pathways to generalizations and unresolved questions.
Abstract
We characterize the permutations of $\mathbb{F}_q$ whose graph minimizes the number of collinear triples and describe the lexicographically-least one, affirming a conjecture of Cooper-Solymosi. This question is closely connected to Dudeney's No-3-in-a-Line problem, the Heilbronn triangle problem, and the structure of finite plane Kakeya sets. We discuss a connection with complete sets of mutually orthogonal latin squares and state a few open problems primarily about general finite affine planes.