Abstract

Let $\alpha :{\mathbb{F}}_q\to {\mathbb{F}}_q$ be a permutation and $\Psi \left(\alpha \right)$ 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 $\Psi \left(\alpha \right)\ge \left(q-1\right)∕4$ and conjectured the sharp bound should be $\Psi \left(\alpha \right)\ge \left(q-1\right)∕2$. In this note we confirm this conjecture.

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Mathematical Subject Classification 2000

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