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 $\Psi (\alpha )\ge (q-1)/4$ and conjectured the sharp bound should be $\Psi (\alpha )\ge (q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true.

4 pages

References in corpus (1)

• On the number of collinear triples in permutations