Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial Vol. 8, No. 1, 2008

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Collinear triples in permutations

Liangpan Li

Vol. 8 (2008), No. 1, 171–173

DOI: 10.2140/iig.2008.8.171

Abstract

Let $α : F_{q} \to F_{q}$ be a permutation and $Ψ (α)$ be the number of collinear triples in the graph of $α$ , where $F_{q}$ denotes a finite field of $q$ elements. When $q$ is odd, Cooper and Solymosi once proved $Ψ (α) \geq (q - 1) ∕ 4$ and conjectured the sharp bound should be $Ψ (α) \geq (q - 1) ∕ 2$ . In this note we confirm this conjecture.

Keywords

collinear triple, permutation, Kakeya set

Mathematical Subject Classification 2000

Primary: 11T99

Milestones

Received: 21 May 2008

Accepted: 2 September 2008

Authors

Liangpan Li