A superlinear improvement on line-free sets in ${\mathbb{F}}_p^3$

Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in ${\mathbb{F}}_p^3$, showing that $r_p({\mathbb{F}}_p^3)\ge (p-1{)}^3+\frac{1}{8}{p}^{3\mathrm{/}2}-O(p)$ as $p\to \mathrm{\infty }$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\dots ,p-2{\}}^3$. By taking the complement of our set, we also get a new upper bound of $3p^2-\frac{1}{8}{p}^{3\mathrm{/}2}+O(p)$ on the smallest size of a $2$-blocking set in the affine geometry $\mathrm{A}\mathrm{G}(3,p)$.