Hindman's theorem does not code ${\mathrm{\varnothing }}^{(\omega )}$ 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.