The extensible no-four-on-a-circle problem

We show that there exists a set $S\subset {\mathbb{Z}}^2$ containing no four points on a circle or a line such that $\mathrm{\mid }S\cap [n{]}^2\mathrm{\mid }=\mathrm{\Omega }(n)$ as $n\to \mathrm{\infty }$. Since any no-four-on-a-circle set in $[n{]}^2$ has size $O(n)$, this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.