Brown’s lemma in second-order arithmetic

Brown’s lemma states that in every finite coloring of the natural numbers there is a homogeneous piecewise syndetic set. We show that Brown’s lemma is equivalent to IΣ02 over RCA∗0. We show in contrast that (infinite) van der Waerden’s theorem is equivalent to BΣ02 over RCA∗0. We finally consider the finite version of Brown’s lemma and show that it is provable in RCA0 but not in RCA∗0.