Erdős' no-three-in-line density conjecture

Let $S\subseteq\mathbb{N}^2$ contain no three points on a common line. No-three-in-line density conjecture.

$$\liminf_{n\to\infty}\frac{|S\cap[n]^2|}{n}=0.$$

Although finite $n\times n$ configurations have the elementary upper bound $2n$, known constructions have density near $n$; the conjecture predicts that every infinite configuration has arbitrarily sparse initial squares.