On the incompleteness of \((k,n)\)-arcs in Desarguesian planes of order \(q\) where \(n\) divides \(q\) (Q1282314)

A \((k, n)\)-arc in a projective plane is a set of \(k\) points, at most \(n\) on every line. A \((k, n)\)-arc is called complete if it cannot be extended to a \((k + 1,n)\)-arc. The authors investigate the completeness of an \((nq - q + n - \varepsilon,n)\)-arc in the Desarguesian plane of order \(q\) where \(n\) divides \(q.\) It is shown that an \((nq - q + n - \varepsilon, n)\)-arc in \(PG(2, q)\), \(n < q/3,\) is incomplete for \(0 < \varepsilon\leqslant n/2\) and any such arc can therefore be completed to a maximal arc. For \(q = 2n\) they are incomplete for \(0 <\varepsilon<0.381n\) and for \(q = 3n\) they are incomplete for \(0 < \varepsilon <0.476n.\) It is known that for \(q\) odd such arcs do not exist for \(\varepsilon = 0\) and, hence, the authors improve the upper bound on the maximum size of such a \((k, n)\)-arc.