Permutations minimizing the number of collinear triples

TL;DR

This paper characterizes permutations over finite fields that minimize collinear triples, confirming a conjecture and connecting to classical combinatorial problems like No-3-in-a-Line and Kakeya sets.

Contribution

It provides a complete characterization of permutations minimizing collinear triples and identifies the lexicographically-least such permutation, confirming a prior conjecture.

Findings

Characterization of permutations minimizing collinear triples

Confirmation of Cooper-Solymosi's conjecture

Connection to finite affine plane problems

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.