Bounds on Arithmetic Rainbow Ramsey Multiplicities

Abstract: We study a quantitative Ramsey-type problem on 3-term arithmetic progressions: how should the set of integers $[n] = {1, 2, \dots, n}$ be colored using 3 colors in order to maximize the number of rainbow 3-term arithmetic progressions? By "rainbow", we mean progressions whose elements are each assigned a distinct color. We determine a lower bound for this question and upper and lower bounds when $[n]$ is replaced with the integers modulo $n$, including an exact maximum when $n$ is a multiple of 3.