From Schur's Theorem to the Pairwise Sum Theorem

We show that by only assuming Schur’s Theorem and the existence of a non- principal ultrafilter, one can directly prove that in every finite coloring of N there exist infinite disjoint sets A,B such that all elements of A∪B∪(A+B) are monochromatic. This gives a partial answer to a question posed by N. Hindman, I. Leader, and D. Strauss in 2003. In the last section we propose a formalization of that open question in purely topological terms.