Canonical theorems for convex sets (Q1387854)

This beautiful paper discusses a theorem that generalizes the famous Erdős-Szekeres theorem (for every \(r \geq 3\), there exists an integer \(f(r)\) such that any set of at least \(f(r)\) points in the plane has \(r\) elements as vertices of a convex \(r\)-gon). We say that a collection of convex sets are in convex position if none of its members is contained in the convex hull of the union of the others. In the present paper the main result says: For any family \(F\) of \(n\) pairwise disjoint compact sets in the plane, such that none of them is contained in the convex hull of two others, and \(r\) a positive integer there are \(r\) disjoint subfamilies \(F_i\) (\(1 \leq i\geq r\)) of cardinality \(c_rn\) (\(c_r\) is a constant that depends on \(r\)) such that no matter how we pick an element \(p_i\) from each \(F_i\) , they are in convex position (as convex sets). In other words, every \(F_i\) appears on the boundary of the convex hull of \(\bigcup_{i=1}^{r} F_i\). The authors present two other Ramsey-type theorems.