The Rado Multiplicity Problem in Vector Spaces over Finite Fields

Abstract

We study an analogue of the Ramsey multiplicity problem for additive structures, establishing the minimum number of monochromatic $3$-APs in $3$-colorings of $\mathbb{F}_3^n$ and obtaining the first non-trivial lower bound for the minimum number of monochromatic $4$-APs in $2$-colorings of $\mathbb{F}_5^n$. The former parallels results by Cumings et al.~\cite{CummingsEtAl_2013} in extremal graph theory and the latter improves upon results of Saad and Wolf~\cite{SaadWolf_2017}. Lower bounds are notably obtained by extending the flag algebra calculus of Razborov~\cite{razborov2007flag}.