A note on the extensible no-three-in-line problem

We show the existence of a set $S\subset\mathbb{Z}^2$ avoiding collinear triples satisfying $|S\cap [n]^2|=\Omega(n/\sqrt{\log n})$ for sufficiently large $n$. This improves on the best-known lower bound on Erde's extensible no-three-in-line problem due to Nagy, Nagy and Woodroofe by $\sqrt{\log n}$, leaving the same gap to the trivial upper bound. Our construction is random.