  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 $\mathbb{F}_3^n$ as well as 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
(2013) in extremal graph theory and the latter improves upon results of Saad
and Wolf (2017) The lower bounds are notably obtained by extending the flag
algebra calculus of Razborov (2007) to additive structures in vector spaces
over finite fields.
