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 $\mathrm{\mid }S\cap [n{]}^2\mathrm{\mid }=\mathrm{\Omega }(n\mathrm{/}\sqrt{\log n})$ 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.