Extending Arcs: An Elementary Proof

Abstract

In a finite projective plane $\pi$ we consider two configuration conditions involving arcs in $\pi$ and show via combinatorial means that they are equivalent. When the conditions hold we are able to obtain embeddability results for arcs, all proofs being elementary. In particular, when $\pi=PG(2,q)$ with $q$ even we provide short proofs of some well known embeddability results.