Maximal line-free sets in ${\mathbb{F}}_p^n$

Abstract

We study subsets of ${\mathbb{F}}_p^n$ that do not contain progressions of length $k$. We denote by $r_k({\mathbb{F}}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and therefore sets containing no full line. A~trivial lower bound $r_p({\mathbb{F}}_p^n)\ge (p-1{)}^n$ is achieved by a hypercube of side length $p-1$ and it is known that equality holds for $n\in \{1,2\}$. We will however show that $r_p({\mathbb{F}}_p^3)\ge (p-1{)}^3+p-2\sqrt{p}$, which is the first improvement in the three dimensional case that is increasing in $p$. We will also give the upper bound $r_p({\mathbb{F}}_p^3)\le p^3-2p^2-(\sqrt{2}-1)p+2$ as well as generalizations for higher dimensions. Finally we present some bounds for individual $p$ and $n$, in particular $r_5({\mathbb{F}}_5^3)\ge 70$ and $r_7({\mathbb{F}}_7^3)\ge 225$ which can be used to give the asymptotic lower bound $4.121^n$ for $r_5({\mathbb{F}}_5^n)$ and $6.082^n$ for $r_7({\mathbb{F}}_7^n)$.