  We show that the problem of counting collinear points in a permutation
(previously considered by the author and J. Solymosi in "Collinear Points in
Permutations", 2005) 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 (5q/14 + O(1)) for the number of
collinear triples of a permutation of GF(q) and a new lower bound (q(q + 1)/2 +
5q/14 + O(1)) on the size of the smallest Besicovitch set in GF(q)^2. Several
interesting questions about the structure of the collinear triple hypergraph
are presented.
