Thin set theorem for arbitrarily many colors implies bounding

Abstract: The thin set theorem $\mathsf{RT}<em>{&lt;\infty,\ell}<sup>{n}$ asserts that for every natural number $k$, each coloring $$c\colon[\mathbb{N}]ⁿ \to {0,1,\dots,k-1}$$ admits an infinite set $H$ such that $|c([H]^n)| \le \ell$. Within the framework of the reverse mathematics of second-order arithmetic, $\mathsf{RT}</em>{&lt;\infty,\ell}<sup>{n}$ implies the $Σ<em>{n+1}<sup>{0}$-bounding principle ($\mathsf{B}Σ</em>{n+1}<sup>{0}$) over $\mathsf{RCA}_0$ for all natural numbers $n, \ell \ge 1$.