Arcs and k-sets with large nucleus-set in Hall planes (Q1118811)

Let K be a set of k points in a projective plane \(\pi_ q\). A point \(P\in K\) is called an internal nucleus of K if every line through P meets K in at most two points. The purpose of this paper is to construct k-sets \((k\geq q+1)\) in the Hall plane H(q) possessing q-\(\sqrt{q}\) internal nuclei and a \((q+1)\)-set in H(q) (q odd) possessing \(q-2\sqrt{q}-1\) internal nuclei. As a by-product complete (q-1)-arcs of H(q) (q odd) and complete \((q-\sqrt{q}+2)-\) arcs of H(q) are constructed.