Uncovered hull vertices and upper bounds for convex polygons

We study points that belong to no convex polygon of a prescribed size. Let Bn be the largest size of a planar point set with no convex (n + 1) polygon and with a hull vertex on no convex n polygon. We give a self contained derivation of ES(n+1) ≤ 3Bn−3 for n ≥ 4. The proof combines a projective correspondence with the classical cup and cap partition. Thus a certified exclusion of 28 points with an uncovered hull vertex at size six would imply ES(7) ≤ 78. The corresponding search uses 27 projected points. We also relate adjacent uncovered hull vertices to Bn and prove restrictions on how the hull changes under deletion. Observed profiles from generated configurations motivate the computational targets. They are sample bounds and do not establish an extremal envelope. The full exclusion at 28 points remains open in our experiments.