On some extremum problems in elementary geometry

1. Let S denote a set of points in the plane, N(S) the number of points in S. More than 25 years ago we have proved [2] the following conjecture of ESTHER KLEINSZERERES: There exists a positive integer f (n) with the property that if iV (S) > f(n) then S contains a subset P with N(P) = n such that the points of P form a convex n-gon. Moreover we have shown that if lo(n) is the smallest such integer then