On the general position subset selection problem

TL;DR

This paper investigates the size of large subsets with no three collinear points within larger point sets constrained by collinearity, providing new bounds that improve upon previous results especially when collinearity bounds grow with the set size.

Contribution

It establishes new lower bounds for the maximum size of subsets with no three collinear points in sets with bounded collinearity, extending and improving prior results.

Findings

For $ ext{l} \, ext{O}(\, ext{sqrt}(n))$, $f(n, ext{l}) \, ext{is} \, ext{at least} \, ext{Omega}( ext{sqrt}(rac{n}{ ext{ln} ext{l}}))$.

For $ ext{l} \, ext{O}(n^{(1- ext{epsilon})/2})$, $f(n, ext{l}) \, ext{is} \, ext{at least} \, ext{Omega}( ext{sqrt}(n ext{log}_ ext{l} n))$.

Provides bounds that improve previous results when $ ext{l}$ is not fixed.

Abstract

Let $f(n,\ell)$ be the maximum integer such that every set of $n$ points in the plane with at most $\ell$ collinear contains a subset of $f(n,\ell)$ points with no three collinear. First we prove that if $\ell \leq O(\sqrt{n})$ then $f(n,\ell)\geq \Omega(\sqrt{\frac{n}{\ln \ell}})$. Second we prove that if $\ell \leq O(n^{(1-\epsilon)/2})$ then $f(n,\ell) \geq \Omega(\sqrt{n\log_\ell n})$, which implies all previously known lower bounds on $f(n,\ell)$ and improves them when $\ell$ is not fixed. A more general problem is to consider subsets with at most $k$ collinear points in a point set with at most $\ell$ collinear. We also prove analogous results in this setting.