Permutations minimizing the number of collinear triples
Joshua Cooper, Jack Hyatt · Designs Codes and Cryptography · 2025
Abstract We characterize the permutations of $$\mathbb {F}_q$$ F q whose graph minimizes the number of collinear triples and describe the lexicographically-least one, confirming a conjecture of Cooper-Solymosi. This question is 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.