Big line or big convex polygon

Let $E{S}_{\mathrm{\ell }}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\mathrm{\ell }$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\mathrm{\ell },n\ge 3$, $$(3\mathrm{\ell }-1)\cdot 2^{n-5}<E{S}_{\mathrm{\ell }}(n)<{\mathrm{\ell }}^2\cdot 2^{n+C\sqrt{n\log n}}\mathrm{.}$$ A similar extension of the well-known Erd\H os–Szekeres cups-caps theorem is also proved.