The Rado multiplicity problem in vector spaces over finite fields

Abstract

We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of F3n as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic 4-APs in 2-colorings of F5n. The former parallels results by Cumings et al. [8] in extremal graph theory and the latter improves upon results of Saad and Wolf [42]. The lower bounds are notably obtained by extending the flag algebra calculus of Razborov [39] to additive structures in vector spaces over finite fields. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.