A combinatorial proof of the Dense Hindman’s Theorem

The Dense Hindman’s Theorem states that, in any finite coloring of the natural numbers, one may find a single color and a “dense” set B₁, for each b₁∈B₁ a “dense” set (depending on b₁), for each a “dense” set (depending on b₁,b₂), and so on, such that for any such sequence of b_i, all finite sums belong to the chosen color. (Here density is often taken to be “piecewise syndetic”, but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.