Big line or big convex polygon

Let ES_(n) be the minimum N such that every N-element point set in the plane contains either collinear members or n points in convex position. We prove that there is a constant $C>0$ such that, for each , n ≥ 3, (3 - 1) · 2ⁿ⁻⁵ < ES_(n) < ² · 2n+ C√nlog n. A similar extension of the well-known Erd os--Szekeres cups-caps theorem is also proved.