Big line or big convex polygon

TL;DR

This paper establishes exponential bounds on the minimum number of points needed in the plane to guarantee either a large collinear subset or a convex polygon, extending classical combinatorial geometry results.

Contribution

It provides new exponential bounds for the Erdős–Szekeres type problem involving collinearity and convex position, extending the cups-caps theorem.

Findings

Upper and lower bounds for $ES_{ ext{ell}}(n)$ involving exponential functions.

Extension of the Erdős–Szekeres cups-caps theorem to collinearity conditions.

Proof of the existence of a constant $C$ for the bounds.

Abstract

Let $ES_{\ell}(n)$ 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 $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.