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

Abstract

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 $r_p(\mathbb{F}_p^3)\ge (p-1)^3+\frac18 p^{3/2} - O(p)$ as $p\to \infty$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\ldots,p-2\}^3$. By taking the complement of our set, we also get a new upper bound of $3p^2-\frac18p^{3/2}+O(p)$ on the smallest size of a $2$-blocking set in the affine geometry $\mathrm{AG}(3,p)$.