# On the number of points in general position in the plane

## TL;DR

This paper investigates the maximum size of point subsets in the plane avoiding four collinear points, and introduces new geometric constructions using the Hypergraph Container Method to study epsilon-nets.

## Contribution

It provides the first known construction of large point sets with no four collinear points that still contain many collinear triples in large subsets, and applies the Hypergraph Container Method to geometric problems.

## Key findings

- Existence of large point sets with no four collinear points but many collinear triples in large subsets
- New geometric constructions using the Hypergraph Container Method
- Results on epsilon-nets in planar point-line systems

## Abstract

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $ it contains a collinear triple. Another application studies epsilon-nets in a point-line system in the plane. We prove the existence of some geometric constructions with a new tool, the so-called Hypergraph Container Method.