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 \(r_7(\mathbb {F}_7^{n})\) .
