Work

Permutations minimizing the number of collinear triples

Year: 2025

Type: article

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.

Source: Designs Codes and Cryptography

DOI: 10.1007/s10623-025-01632-w

Authors: Joshua Cooper, Jack Hyatt

Institution: University of South Carolina

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funder: University of South Carolina