  We study a 3D ternary system derived as a sharp-interface limit of the
Nakazawa-Ohta density functional theory of triblock copolymers, which combines
an interface energy with a long range interaction term. Although both the
binary case in 2D and 3D, and the ternary case in 2D, are quite well studied,
very little is known about the ternary case in 3D. In particular, it is even
unclear whether minimizers are made of finitely many components. In this paper
we provide a positive answer to this, by proving that the number of components
in a minimizer is bounded from above by some quantity depending only on the
total masses and the interaction coefficients. One key difficulty is that the
3D structure prevents us from uncoupling the Coulomb-like long range
interaction from the perimeter term, hence the actual shape of minimizers is
unknown, not even for small masses. This is due to the lack of a quantitative
isoperimetric inequality with two mass constraints in 3D, and it makes the
construction of competitors significantly more delicate.
