On the embedding of \((k,p)\)-arcs in maximal arcs (Q1591646)

A \((k,n)\)-arc in a projective plane of order \(q\) is a set of \(k\) points no \(n+1\) on a line. The author proves that every \((k,p)\)-arc in \(\text{PG}(2,q)\), \(q=p^h\), \(p\) prime and \(k \geq qp-q+p-\frac{1}{2}{\root 4\of q}\), can be extended to a maximal arc. Using the nonexistence of maximal arcs for \(p > 2\) [\textit{S. Ball, A. Blokhuis} and \textit{F. Mazzocca}, Combinatorica 17, No. 1, 31-41 (1997; Zbl 0880.51003)], an upper bound for the size of a \((k,p)\)-arc is obtained.