k-arcs in A Projective Plane
In a finite projective plane π (not necessarily Desarguesian) a set A of k (k ≥ 3) points such that no three points of A are collinear (on a line) is called a k-arc. If the plane π has order q then k ≤ q + 2, however the maximum value of k can only be achieved if q is even. In a plane of order q, a (q + 1)-arc is called an oval and, if q is even, a (q + 2)-arc is called a hyperoval.
A k-arc which can not be extended to a larger arc is called a complete arc. In the Desarguesian projective planes, PG(2,q), no q-arc is complete, so they may all be extended to ovals.
Read more about this topic: Arc (projective Geometry)
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