Collinear triples in permutations

TL;DR

This paper proves the conjecture by Cooper and Solymosi that the minimum number of collinear triples in the graph of a permutation over a finite field is at least half of the field size minus one.

Contribution

It confirms the sharp lower bound conjectured for collinear triples in permutation graphs over finite fields.

Findings

Proves the Cooper-Solymosi conjecture.

Establishes the exact minimum number of collinear triples.

Provides insight into geometric configurations over finite fields.

Abstract

Let $\alpha:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples in the graph of $\alpha$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once proved $\Psi(\alpha)\geq(q-1)/4$ and conjectured the sharp bound should be $\Psi(\alpha)\geq(q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true.