Hindman's theorem does not code $\emptyset^{(\omega)}$ in one application

We prove that for every non-arithmetic set~$C$ and every arithmetic finite coloring of~$\mathbb{N}$, there is an infinite set $H \subseteq \mathbb{N}$ whose non-empty finite sums of distinct elements is monochromatic, and $C$ is not $H$-computable.

We also study restrictions of Hindman's theorem to simple colorings.