Collinear triples in permutations

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.