Thin set theorem for arbitrarily many colors implies bounding

Abstract

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

10 pages, 1 figure