Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem

We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in [1]) and the well-known finite plane Kakeya problem are intimately connected. Via counting arguments and by studying the hypergraph of collinear triples we show a new lower bound (25q/72 + O(1)) for the number of collinear triples of a permutation of Fq and a new lower bound (q(q + 1)/2 + 25q/72 + O(1)) on the size of the smallest Besicovitch set in Fq. Several interesting questions about the structure of the collinear triple hypergraph are presented.