András Bezdek - A Sylwester type theorem on circles

In spite of the specific title, generalizations of two combinatorial geometric problems will be disscussed. The classical result of Sylvester shows that n points in the plane, not all on a common line, there exists an ordinary line (a line through exactly two points). The similar question about unit circles will be disscussed. A problem of V. Boltyanski (proved by A. V. Bogomolnaya, A. V. Nazarov and S. Rukshin) claims that if n points in a convex n-sided polygon are given, then using the sides and the given points n side-point pairs can be formed such that the triangles determined by the pairs cover the convex polygon. We strengthen this result by proving that the assumption according to which the given points are inside of the polygon is redundant. We also give a bound for the number of disjoint such triangles.