Sets in Almost General Position

TL;DR

This paper investigates the size of large subsets in almost general position within point sets in various dimensions, providing new bounds and answering longstanding questions about the existence of large general position subsets.

Contribution

It extends Erd ext{"o}s' question to higher dimensions, offering new bounds and solutions for the size of guaranteed general position subsets in almost general position point sets.

Findings

Established that in all dimensions, the largest guaranteed subset in general position can be smaller than linear in n.

Extended results from 2D and 3D to higher dimensions, answering open questions.

Derived improved bounds for the size of such subsets in certain dimensions.

Abstract

Erd\H{o}s asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? F\"uredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, T\'oth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in general position we can guarantee to find has size $o(n)$? We answer their question for all $d$ and derive improved bounds for certain dimensions.