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) 2^n-5 < ES_ (n) < ² 2^n+ Cn n. A similar extension of the well-known Erd os--Szekeres cups-caps theorem is also proved.