Lower Bounds for Complete {k; n}-ARCS

Let rc be a finite-projective plane. Then a (k; n)-arc is a set K of k points of IC such that no more than n of them are collinear. K is said to be complete if K is not properly contained in any other (k; n)-arc. In accordance with the standard notation the order of R is denoted by q (whether or not q is a prime power). In his paper [l] where such arcs are studied Barlotti points out that k < nq + n q. If k = nq + n q, then K is called maximal. These maximal arcs have been extensively studied in [3, 71, and in more recent papers of Hill and Mason [4] and Wilson [6]. In (21 Barnabei et al. obtain a lower bound for the cardinality of a complete {k; q}-arc in a plane of order q*. Their proof is ingenious but intricate. In this note we shall generalize that main result of [2] and our proof is considerably less complicated. We also obtain a characterization of Baer subplanes as complete {k; q + 1 }-arcs of minimum cardinality in a plane of order q*.