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

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

collinear triple, permutation, Kakeya set
Mathematical Subject Classification 2000
Primary: 11T99
Received: 21 May 2008
Accepted: 2 September 2008
Liangpan Li