General position subsets

Importance: Medium ✭✭
Author(s): Gowers, Timothy
Subject: Geometry
Recomm. for undergrads: no
Posted by: David Wood
on: March 24th, 2014

\begin{question} What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ points in general position (no three collinear)? \end{question}

The $n\times n$ grid contains no set of $n+1$ collinear points and no subset of $2n+1$ points in general position, implying $f(n)\geq \Omega(n^2)$.

To see that $f(n)\leq O(n^3)$, consider a set $P$ of points that contain no $n$ collinear points, and contain no subset of $n$ points in general position. Let $S$ be a maximal subset of $P$ in general position. Every point in $P-S$ is on one of the $\binom{|S|}{2}$ lines determined by $S$. Each such line contains at most $n-3$ points in $P-S$. Thus $|P|\leq |S|+\binom{|S|}{2}(n-3) \leq (n-1)+\binom{n-1}{2}(n-3)\leq O(n^3)$.

Payne and Wood [PW] improved this upper bound to $f(n)\leq O(n^2\log n)$. The proof is based on the Szemerédi-Trotter Theorem and Spencer's Lemma about independent sets in hypergraphs.

It is reasonable to think that the grid is the extremal example, and $f(n)\leq O(n^2)$. This would be an elegant generalisation of a result by Erdős [R] who proved that the $n\times n$ grid contains a subset of $n-o(n)$ points in general position (the \href[no-three-in-line problem]{http://en.wikipedia.org/wiki/No-three-in-line_problem}).

Bibliography

*[G] Timothy Gowers, \href[A geometric Ramsey problem]{http://mathoverflow.net/questions/50928/a-geometric-ramsey-problem}.

[PW] Michael Payne, David R. Wood. On the general position subset selection problem, SIAM J. Discrete Math. 27.4:1727-1733, 2013.

[R] K. F. Roth, On a problem of Heilbronn, J. London Mathematical Society 26.3:198–204, 1951.


* indicates original appearance(s) of problem.