Note on (q+2)-Sets in A Galois Plane of Order q

Let Ω be a (q+2)-set in a projective plane PG(2, q) and let ϕ ⊆ Ω be the set of the points of Ω with the property that every line containing a point of ϕ intersects Ω in two points. In this paper we prove that: If |ϕ gt; 2, then q is even; if |ϕ > q÷2, then ϕ = Ω for each q even, there exists an Ω such that |ϕ = q÷2.