Big line or big convex polygon

Abstract

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