Nine convex sets determine a pentagon with convex sets as vertices (Q1123407)

The paper deals with a generalization of a famous Erdős-Szekeres problem about convexly independent points. Let \({\mathcal F}\) be a family of mutually disjoint ovals (i.e. compact convex subsets) in the plane. Any three ovals from \({\mathcal F}\) are called collinear if the convex hull of two of them contains the third. Some ovals \(A_ 1,...,A_ n\) from \({\mathcal F}\) form an n-gon provided \(A_ i\not\subset conv\{UA_ j:\) \(j\neq i\}\), \(i=1,...,n.\) Theorem. If \(| {\mathcal F}| >8\), then some five ovals from \({\mathcal F}\) determine a 5-gon.