A Lower Bound for the Number of Polygonizations of \(N\) Points in the Plane

Let \(\Phi(N)\) be the maximum number of simple polygons that can be drawn using as vertices a set \(V\) of \(N\) points in the plane. By counting the number of simple polygons of a particular configuration of \(V\), an improved lower bound for \(\Phi(N)\) is obtained. It is proved that \(\Phi(N)^\frac{1}{N}\) is asymptotically greater than \(3.6\).