Asymptotic minimum under exact interval colorings

Determine, as a function of δ, the asymptotic minimum number of monochromatic and rainbow solutions to x + y < z with x ≤ y over all δn-exact 3-colorings of [1,n] whose color classes are single intervals.

Background

The set D consists of 3-colorings of [1,n] in which each color class is a single interval, and Dδ consists of those colorings in D that use each color at least δn times. The paper derives boundary descriptions and an example for δ=0.1, but does not obtain the general asymptotic minimum as a function of δ. This unresolved optimization problem is stated explicitly among the open questions.