Bounds for Gallai-Schur triples in k-exact r-colorings

Determine, for every r ≥ 3 and k-exact r-colorings of [1,n], asymptotic lower and upper bounds on the minimum number of Gallai-Schur triples, and determine the conclusion when k = δn with 0 < δ < 1/r.

Background

The paper defines a k-exact coloring as an r-coloring in which every color is used at least k times. Theorem 4.3 gives bounds for the minimum number of Gallai-Schur triples over unrestricted r-colorings, but its upper bound is independent of r and the authors note that exact coloring may not be the appropriate constraint for this multiplicity problem. They therefore leave open the determination of bounds under the stronger k-exactness requirement, including the linear regime k=δn.