2023/10/05 by Christian Elsholtz, Elsholtz, Christian, Jakob Führer +11
Mathematics · #05D05 #11B25 #51E21 #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2310.03382
openalex publication_date 2023/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study subsets of \mathbbFpn that do not contain progressions of length k. We denote by rk(\mathbbFpn) 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 rp(\mathbbFpn)≥(p-1)n is achieved by a hypercube of side length p-1 and it is known that equality holds for n∈\1,2\. We will however show that rp(\mathbbFp3)≥ (p-1)3+p-2√(p), which is the first improvement in the three dimensional case that is increasing in p. We will also give the upper bound rp(\mathbbFp3)≤ p3-2p2-(√(2)-1)p+2 as well as generalizations for higher dimensions. Finally we present some bounds for individual p and n, in particular r5(\mathbbF53)≥ 70 and r7(\mathbbF73)≥ 225 which can be used to give the asymptotic lower bound 4.121n for r5(\mathbbF5n) and 6.082n for r7(\mathbbF7n).