Big line or big convex polygon

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