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 $r_p({\mathbb{F}}_p^3)\ge (p-1{)}^3+\frac{1}{8}{p}^{3/2}-O(p)$ as $p\to \infty$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\dots ,p-2{\}}^3$. Via complementation, a line-free set in ${\mathbb{F}}_p^3$ corresponds to a $2$-blocking set in the affine geometry $AG(3,p)$, hence we also obtain an upper bound of $3p^2-\frac{1}{8}{p}^{3/2}+O(p)$ on the smallest size of such a $2$-blocking set.

arXiv comment7 pages, 1 figure