  The thin set theorem for $n$-tuples and $k$ colors ($\mathsf{TS}^n_k$) states
that every $k$-coloring of $[\mathbb{N}]^n$ admits an infinite set of integers
$H$ such that $[H]^n$ avoids at least one color. In this paper, we study the
combinatorial weakness of the thin set theorem in reverse mathematics by
proving neither $\mathsf{TS}^n_k$, nor the free set theorem ($\mathsf{FS}^n$)
imply the Erd\H{o}s-Moser theorem ($\mathsf{EM}$) whenever $k$ is sufficiently
large. Given a problem $\mathsf{P}$, a computable instance of $\mathsf{P}$ is
universal iff its solution computes a solution of any other computable
$\mathsf{P}$-instance. We prove that Erd\H{o}s-Moser theorem does not admit a
universal instance.
