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

• Published in: Periodica mathematica Hungarica 2025-01
• Main Authors: Elsholtz, Christian, Führer, Jakob, Füredi, Erik, Kovács, Benedek, Pach, Péter Pál, Simon, Dániel Gábor, Velich, Nóra
• Format: Article
• Language: English
• ISSN: 0031-5303; 1588-2829
• DOI: 10.1007/s10998-024-00617-x

We study subsets of $$\mathbb {F}_p^n$$ F p n that do not contain progressions of length $$k$$ k . We denote by $$r_k(\mathbb {F}_p^n)$$ r k ( F p n ) the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $$k=p$$ k = p and therefore sets containing no full line. A trivial lower bound $$r_p(\mathbb {F}_p^n)\ge (p-1)^n$$ r p ( F p n ) ≥ ( p - 1 ) n is achieved by a hypercube of side length $$p-1$$ p - 1 and it is known that equality holds for $$n\in \{1,2\}$$ n ∈ { 1 , 2 } . We will however show that $$r_p(\mathbb {F}_p^3)\ge (p-1)^3+p-2\sqrt{p}$$ r p ( F p 3 ) ≥ ( p - 1 ) 3 + p - 2 p , which is the first improvement in the three-dimensional case that is increasing in $$p$$ p . We will also give the upper bound $$r_p(\mathbb {F}_p^{3})\le p^3-2p^2-(\sqrt{2}-1)p+2$$ r p ( F p 3 ) ≤ p 3 - 2 p 2 - ( 2 - 1 ) p + 2 as well as generalizations for higher dimensions. Finally, we present some bounds for individual $$p$$ p and $$n$$ n , in particular $$r_5(\mathbb {F}_5^{3})\ge 70$$ r 5 ( F 5 3 ) ≥ 70 and $$r_7(\mathbb {F}_7^{3})\ge 225$$ r 7 ( F 7 3 ) ≥ 225 which can be used to give the asymptotic lower bound $$4.121^n$$ 4 . 121 n for $$r_5(\mathbb {F}_5^{n})$$ r 5 ( F 5 n ) and $$6.082^n$$ 6 . 082 n for $$r_7(\mathbb {F}_7^{n})$$ r 7 ( F 7 n ) .