A Kakeya set $\mathcal{ K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{ L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with size at least $q^2-3q+9$ contain a large knot (a point of $\mathcal{ K}$ lying on many lines of $\mathcal{ L}$). In this paper, we improve on this result by showing that Kakeya set of size at least $\approx q^2-q\sqrt{ q}+\frac{ 3}{ 2}q$ contain a large knot. Furthermore, we obtain a sharp result for planes of square order containing a Baer subplane.
Categories: math.CO